MacBook Pro 13" (Mid-2012, A1278) Fan making loud noise.

If your laptop is the non-Retina model, then you can still get it serviced by an Apple Store or an Apple Authorized Service Provider. If it is a Retina model, then it may still be serviceable by Apple even though it is considered Vintage depending on whether Apple still has the fan in stock in their parts warehouse (this is a relatively new option for some Vintage units and may vary by region).

Check to make sure no foreign object is stuck in the fan and make sure the fan housing is not bent which could interfere with the fan blades. Is the Bottom Case dented below where the fan sits?

If the fan bearing is failing, then you may be able to find a replacement online if you don't want Apple to service the unit (or Apple is unable to procure the part). An independent Service Provider may be able to acquire the fan and replace it for you).

It is also possible the heatsink thermal compound is dried up and needs replaced which can help lower the temperature of the CPU.

This happened to my old MacBook that I used as a server. It held up really well, but the fan ended up failing after 3-4 years of continuously running. I'd take this to be a fairly good lifespan, especially being used way more than it should. The fan still spins, but very slowly. I was able to improve it slightly by removing the fan and blasting it with compressed air. This didn't fully remedy it either. If you're really experience, you could try taking the fan itself apart to see if there is any debris wrapped around it. If not, you may be able to buy a new fan from Apple for $40-$50 bucks.

Be really careful running your Mac without a fan though.

Keep an eye on the temperatures. Anything over 150 degrees Fahrenheit (65 Celcius) sustained for a few hours could cause damage to the expensive components in your laptop. I'd be much better to bite the bullet now and get the fans replaced then to have to replace the $700+ repair for a fried CPU.