Here is the beginnings of something you can use:
browser.storage.local.get(null)
    .then(store => {
            var bytes = 0;
            for (var p in store) {
                var entry = store[p];
                switch (typeof(p)) {
                    case 'number':
                        bytes += 8;
                        break;
                    case 'boolean':
                        bytes += 4;
                        break;
                    case 'string':
                        bytes += (entry.length * 2);
                        break;
                    case 'undefined':
                        bytes += 0;
                        break;
                    case 'object':
                        if (entry === null) {
                            bytes += 0;
                        } else {
                            if (entry.byteLength) {
                                bytes += entry.byteLength;
                            } else if (entry.buffer) {
                                bytes += entry.buffer.byteLength;
                            } else {
                                // todo: recrusively run this on all entries
                            }
                        }
                        break;
                    case 'array':
                        // todo: recursively run this on all elements
                        break;
                    default:
                        if (
                        }
                }
            });
        .catch(ex => console.warn('run borwser.storage.local.get again as it hit error, ex:', ex));
solutions here provide some recursive things - https://stackoverflow.com/a/6351386/1828637