This question is old, the other answers are old. "Employed Russian"s answer is very good and informative, but it only works if you have the source code. If you don't, the alternatives back then were very tricky. Fortunately nowadays we have a simple solution to this problem (as commented in one of his replies), using patchelf. All you have to do is:
$ ./patchelf --set-interpreter /path/to/newglibc/ld-linux.so.2 --set-rpath /path/to/newglibc/ myapp
And after that, you can just execute your file:
$ ./myapp
No need to chroot or manually edit binaries, thankfully. But remember to backup your binary before patching it, if you're not sure what you're doing, because it modifies your binary file. After you patch it, you can't restore the old path to interpreter/rpath. If it doesn't work, you'll have to keep patching it until you find the path that will actually work... Well, it doesn't have to be a trial-and-error process. For example, in OP's example, he needed GLIBC_2.3, so you can easily find which lib provides that version using strings:
$ strings /lib/i686/libc.so.6 | grep GLIBC_2.3
$ strings /path/to/newglib/libc.so.6 | grep GLIBC_2.3
In theory, the first grep would come empty because the system libc doesn't have the version he wants, and the 2nd one should output GLIBC_2.3 because it has the version myapp is using, so we know we can patchelf our binary using that path. If you get a segmentation fault, read the note at the end.
When you try to run a binary in linux, the binary tries to load the linker, then the libraries, and they should all be in the path and/or in the right place. If your problem is with the linker and you want to find out which path your binary is looking for, you can find out with this command:
$ readelf -l myapp | grep interpreter
[Requesting program interpreter: /lib/ld-linux.so.2]
If your problem is with the libs, commands that will give you the libs being used are:
$ readelf -d myapp | grep Shared
$ ldd myapp
This will list the libs that your binary needs, but you probably already know the problematic ones, since they are already yielding errors as in OP's case.
"patchelf" works for many different problems that you may encounter while trying to run a program, related to these 2 problems. For example, if you get: ELF file OS ABI invalid, it may be fixed by setting a new loader (the --set-interpreter part of the command) as I explain here. Another example is for the problem of getting No such file or directory when you run a file that is there and executable, as exemplified here. In that particular case, OP was missing a link to the loader, but maybe in your case you don't have root access and can't create the link. Setting a new interpreter would solve your problem.
Thanks Employed Russian and Michael Pankov for the insight and solution!
Note for segmentation fault: you might be in the case where myapp uses several libs, and most of them are ok but some are not; then you patchelf it to a new dir, and you get segmentation fault. When you patchelf your binary, you change the path of several libs, even if some were originally in a different path. Take a look at my example below:
$ ldd myapp
./myapp: /usr/lib/x86_64-linux-gnu/libstdc++.so.6: version `GLIBCXX_3.4.20' not found (required by ./myapp)
./myapp: /usr/lib/x86_64-linux-gnu/libstdc++.so.6: version `GLIBCXX_3.4.21' not found (required by ./myapp)
linux-vdso.so.1 => (0x00007fffb167c000)
libm.so.6 => /lib/x86_64-linux-gnu/libm.so.6 (0x00007f9a9aad2000)
libdl.so.2 => /lib/x86_64-linux-gnu/libdl.so.2 (0x00007f9a9a8ce000)
libpthread.so.0 => /lib/x86_64-linux-gnu/libpthread.so.0 (0x00007f9a9a6af000)
libstdc++.so.6 => /usr/lib/x86_64-linux-gnu/libstdc++.so.6 (0x00007f9a9a3ab000)
libc.so.6 => /lib/x86_64-linux-gnu/libc.so.6 (0x00007f9a99fe6000)
/lib64/ld-linux-x86-64.so.2 (0x00007f9a9adeb000)
libgcc_s.so.1 => /lib/x86_64-linux-gnu/libgcc_s.so.1 (0x00007f9a99dcf000)
Note that most libs are in /lib/x86_64-linux-gnu/ but the problematic one (libstdc++.so.6) is on /usr/lib/x86_64-linux-gnu. After I patchelf'ed myapp to point to /path/to/mylibs, I got segmentation fault. For some reason, the libs are not totally compatible with the binary. Since myapp didn't complain about the original libs, I copied them from /lib/x86_64-linux-gnu/ to /path/to/mylibs2, and I also copied libstdc++.so.6 from /path/to/mylibs there. Then I patchelf'ed it to /path/to/mylibs2, and myapp works now. If your binary uses different libs, and you have different versions, it might happen that you can't fix your situation. :( But if it's possible, mixing libs might be the way. It's not ideal, but maybe it will work. Good luck!