Is there any way to open or convert an old Oracle 7.x database into Oracle 11g?
Thanks in advance!
No, you have to upgrade in stages, which wouldn't actually be possible now because Oracle 8.x and 9.x are now unsupported and the software is unavailable (unless you have an Oracle support contract and ask them for copies of the software). Read the Oracle® Database Upgrade Guide 11g Release 2 (11.2) for more information.
Your best bet, assuming you have the 7.x database up and running, is to use
exp to export the schemas you require, then import them into a fresh 11.2.x database.