Is anyone aware of a way to use APC on a 1and1 shared hosting account? They don't have the pecl extension installed by default.
It's not APC but you still can setup Memcached on 1and1 shared hosting. see http://wiki.theplaz.com/Memcached_on_1and1_with_MediaWiki I did so myself (for a mediawiki) and it worked quite well.
I doubt they will allow you to install or use APC on shared hosting, as it will affect every site on the server. This could be a potentially bad thing if it causes a conflict or incompatibility with someone else's code. If you want to use APC or something that's not generally offered on shared hosting you'll need either a VPS or dedicated server.