You can use your existing refrigerator, but unfortunately, it is 10-15 degrees cooler than a cave and has a tendency to suck the moisture out of anything that is unprotected.
Make sure it's still within the temperatures recommended on your recipe. Too cold could inhibit mould growth or correct ageing.
To protect the cheese, place it in the warmest part of the refrigerator in an airtight container to keep it from drying out.
The size of the container should be much larger than the cheese, ie 40% cheese and 60% empty space. A container will also help keep any other bacteria and contamination away from the cheese.
You can control the humidity of the air inside the container by using a wet paper towel, crumpled up in a ball and placed in a corner of the container. Remove it if condensation is dripping down the sides of the container. Too much humidity could inhibit correct ageing.
Another option is to use a wine fridge. This will give perfect temperature control for ageing your cheese and you can separate it from other foods to more easily avoid introduction of other bacteria.