Is there a database somewhere for sharing translations of mathematical works?

Put it on arxiv.org, in the math.HO category ("HO" stands for "History and Overview"). I've seen (and read) contributions there that are no more than translations of Euler from Latin to English. You'll want to write it up like a paper, with an explanatory paragraph providing context before the actual translation. This plan seems to provide everything you want, since arxiv.org is indexed by search engines, available to everyone, and should last a long time.

I do not know of such database, but I like the idea of having one very much. Then we can ask all students passing the language exam to contribute to it:-) Mathematical papers are very rarely translated from French and German to English. (Most papers of Gauss and Riemann are not available in a good English translation). Unlike Russian papers: almost all main Russian journals are translated cover-to-cover, at least this was the case few years ago. Presumably people think that if you understand the alphabet then you can read mathematics written in this alphabet in any language:-)

Meanwhile, before this database is created, you can submit your translation to the arXiv. People do submit translations to the arxiv. Search for example on the name: Euler.