A FileStream will allow you to seek to the portion of the file you want, no problem.  It's the recommended way to do it in C#, and it's fast.
Sharing between threads:  You will need to create a lock to prevent other threads from changing the FileStream position while you're trying to read from it.  The simplest way to do this:
//  This really needs to be a member-level variable;
private static readonly object fsLock = new object();
//  Instantiate this in a static constructor or initialize() method
private static FileStream fs = new FileStream("myFile.txt", FileMode.Open);
public string ReadFile(int fileOffset) {
    byte[] buffer = new byte[bufferSize];
    int arrayOffset = 0;
    lock (fsLock) {
        fs.Seek(fileOffset, SeekOrigin.Begin);
        int numBytesRead = fs.Read(bytes, arrayOffset , bufferSize);
        //  Typically used if you're in a loop, reading blocks at a time
        arrayOffset += numBytesRead;
    }
    // Do what you want to the byte array and return it
}
Add try..catch statements and other code as necessary.  Everywhere you access this FileStream, put a lock on the member-level variable fsLock... this will keep other methods from reading/manipulating the file pointer while you're trying to read.
Speed-wise, I think you'll find you're limited by disk access speeds, not code.
You'll have to think through all the issues about multi-threaded file access... who intializes/opens the file, who closes it, etc.  There's a lot of ground to cover.