You can use
hardcopy -h command to save the contents of the current scroll buffer
to a file. As described in
hardcopy [-h] [file]
Writes out the currently displayed image to the file file, or,
if no filename is specified, to hardcopy.n in the default
directory, where n is the number of the current window. This
either appends or overwrites the file if it exists. See below.
If the option -h is specified, dump also the contents of the
I estimate it to be below 5000 lines.
5000 lines is really a lot. The default length of scroll buffer in
screen is just 100, not ~5000 lines. Unless you started your
screen session with a larger scroll buffer setting it will not be
possible to retrieve all ~5000 lines of the scroll buffer.