I like having anchovies in some form or another around, and whenever I buy them (in paste or jarred form), I end up with extras and store them in the fridge. Searching around on the Web has a mix of opinions on their shelf life, with some saying days or weeks, and other saying months or years. The longer recommendations attribute the jarred anchovies lasting longer because of the oil ("as long as they're completely covered") and the paste being safe because of the salt, along with metal tubing not allowing air to ruin it.
These seem like reasonable claims, but I'd prefer a bit more information on this. Thanks!