The answer of @CommonsWare is the one I'd use. However, it looks like KitKat is having some bugs when that's done (When you create a CookieManager with custom CookieStore which you need if you want persistent Cookies).
Given the fact that regardless of the implementation of the CookieStore that is used, Volley would throw a NullpointerException, I had to create my own CookieHandler... use it if you find it helpful.
public class MyCookieHandler extends CookieHandler {
private static final String VERSION_ZERO_HEADER = "Set-cookie";
private static final String VERSION_ONE_HEADER = "Set-cookie2";
private static final String COOKIE_HEADER = "Cookie";
private static final String COOKIE_FILE = "Cookies";
private Map<String, Map<String, HttpCookie>> urisMap;
private Context context;
public MyCookieHandler(Context context) {
    this.context = context;
    loadCookies();
}
@SuppressWarnings("unchecked")
private void loadCookies() {
    File file = context.getFileStreamPath(COOKIE_FILE);
    if (file.exists())
        try {
            FileInputStream fis = context.openFileInput(COOKIE_FILE);
            BufferedReader br = new BufferedReader(new InputStreamReader(
                    fis));
            String line = br.readLine();
            StringBuilder sb = new StringBuilder();
            while (line != null) {
                sb.append(line);
                line = br.readLine();
            }
            Log.d("MyCookieHandler.loadCookies", sb.toString());
            JSONObject jsonuris = new JSONObject(sb.toString());
            urisMap = new HashMap<String, Map<String, HttpCookie>>();
            Iterator<String> jsonurisiter = jsonuris.keys();
            while (jsonurisiter.hasNext()) {
                String prop = jsonurisiter.next();
                HashMap<String, HttpCookie> cookiesMap = new HashMap<String, HttpCookie>();
                JSONObject jsoncookies = jsonuris.getJSONObject(prop);
                Iterator<String> jsoncookiesiter = jsoncookies.keys();
                while (jsoncookiesiter.hasNext()) {
                    String pprop = jsoncookiesiter.next();
                    cookiesMap.put(pprop,
                            jsonToCookie(jsoncookies.getJSONObject(pprop)));
                }
                urisMap.put(prop, cookiesMap);
            }
        } catch (Exception e) {
            e.printStackTrace();
        }
    else {
        urisMap = new HashMap<String, Map<String, HttpCookie>>();
    }
}
@Override
public Map<String, List<String>> get(URI arg0,
        Map<String, List<String>> arg1) throws IOException {
    Log.d("MyCookieHandler.get",
            "getting Cookies for domain: " + arg0.getHost());
    Map<String, HttpCookie> cookies = urisMap.get(arg0.getHost());
    if (cookies != null)
        for (Entry<String, HttpCookie> cookie : cookies.entrySet()) {
            if (cookie.getValue().hasExpired()) {
                cookies.remove(cookie.getKey());
            }
        }
    if (cookies == null || cookies.isEmpty()) {
        Log.d("MyCookieHandler.get", "======");
        return Collections.emptyMap();
    }
    Log.d("MyCookieHandler.get",
            "Cookie : " + TextUtils.join("; ", cookies.values()));
    Log.d("MyCookieHandler.get", "======");
    return Collections.singletonMap(COOKIE_HEADER, Collections
            .singletonList(TextUtils.join("; ", cookies.values())));
}
@Override
public void put(URI uri, Map<String, List<String>> arg1) throws IOException {
    Map<String, HttpCookie> cookies = parseCookies(arg1);
    Log.d("MyCookieHandler.put",
            "saving Cookies for domain: " + uri.getHost());
    addCookies(uri, cookies);
    Log.d("MyCookieHandler.put",
            "Cookie : " + TextUtils.join("; ", cookies.values()));
    Log.d("MyCookieHandler.put", "======");
}
private void addCookies(URI uri, Map<String, HttpCookie> cookies) {
    if (!cookies.isEmpty()) {
        if (urisMap.get(uri.getHost()) == null) {
            urisMap.put(uri.getHost(), cookies);
        } else {
            urisMap.get(uri.getHost()).putAll(cookies);
        }
        saveCookies();
    }
}
private void saveCookies() {
    try {
        FileOutputStream fos = context.openFileOutput(COOKIE_FILE,
                Context.MODE_PRIVATE);
        JSONObject jsonuris = new JSONObject();
        for (Entry<String, Map<String, HttpCookie>> uris : urisMap
                .entrySet()) {
            JSONObject jsoncookies = new JSONObject();
            for (Entry<String, HttpCookie> savedCookies : uris.getValue()
                    .entrySet()) {
                jsoncookies.put(savedCookies.getKey(),
                        cookieToJson(savedCookies.getValue()));
            }
            jsonuris.put(uris.getKey(), jsoncookies);
        }
        fos.write(jsonuris.toString().getBytes());
        fos.close();
        Log.d("MyCookieHandler.addCookies", jsonuris.toString());
    } catch (Exception e) {
        e.printStackTrace();
    }
}
private static JSONObject cookieToJson(HttpCookie cookie) {
    JSONObject jsoncookie = new JSONObject();
    try {
        jsoncookie.put("discard", cookie.getDiscard());
        jsoncookie.put("maxAge", cookie.getMaxAge());
        jsoncookie.put("secure", cookie.getSecure());
        jsoncookie.put("version", cookie.getVersion());
        jsoncookie.put("comment", cookie.getComment());
        jsoncookie.put("commentURL", cookie.getCommentURL());
        jsoncookie.put("domain", cookie.getDomain());
        jsoncookie.put("name", cookie.getName());
        jsoncookie.put("path", cookie.getPath());
        jsoncookie.put("portlist", cookie.getPortlist());
        jsoncookie.put("value", cookie.getValue());
    } catch (JSONException e) {
        e.printStackTrace();
    }
    return jsoncookie;
}
private static HttpCookie jsonToCookie(JSONObject jsonObject) {
    HttpCookie httpCookie;
    try {
        httpCookie = new HttpCookie(jsonObject.getString("name"),
                jsonObject.getString("value"));
        if (jsonObject.has("comment"))
            httpCookie.setComment(jsonObject.getString("comment"));
        if (jsonObject.has("commentURL"))
            httpCookie.setCommentURL(jsonObject.getString("commentURL"));
        if (jsonObject.has("discard"))
            httpCookie.setDiscard(jsonObject.getBoolean("discard"));
        if (jsonObject.has("domain"))
            httpCookie.setDomain(jsonObject.getString("domain"));
        if (jsonObject.has("maxAge"))
            httpCookie.setMaxAge(jsonObject.getLong("maxAge"));
        if (jsonObject.has("path"))
            httpCookie.setPath(jsonObject.getString("path"));
        if (jsonObject.has("portlist"))
            httpCookie.setPortlist(jsonObject.getString("portlist"));
        if (jsonObject.has("secure"))
            httpCookie.setSecure(jsonObject.getBoolean("secure"));
        if (jsonObject.has("version"))
            httpCookie.setVersion(jsonObject.getInt("version"));
        return httpCookie;
    } catch (JSONException e) {
        e.printStackTrace();
    }
    return null;
}
private Map<String, HttpCookie> parseCookies(Map<String, List<String>> map) {
    Map<String, HttpCookie> response = new HashMap<String, HttpCookie>();
    for (Entry<String, List<String>> e : map.entrySet()) {
        String key = e.getKey();
        if (key != null
                && (key.equalsIgnoreCase(VERSION_ONE_HEADER) || key
                        .equalsIgnoreCase(VERSION_ZERO_HEADER))) {
            for (String cookie : e.getValue()) {
                try {
                    for (HttpCookie htpc : HttpCookie.parse(cookie)) {
                        response.put(htpc.getName(), htpc);
                    }
                } catch (Exception e1) {
                    Log.e("MyCookieHandler.parseCookies",
                            "Error parsing cookies", e1);
                }
            }
        }
    }
    return response;
}
}
This answer hasn't been thoroughly tested. I used JSON to serialize Cookies, because well, that class don't implement Serializable and it's final.