Is it possible to Pickle meat at room temperature?

Pickling meat and fish was done for millennia before the advent of refrigeration. Pickling and smoking or drying/curing were the only reliable ways to preserve meat before freezing and canning were invented relatively recently.

The problem is that the term "pickling" is a bit ambiguous. It is a generic term that is used to describe preserving with salt. It can refer to curing or lactic acid fermentation depending on the salt concentration.

When meat or fish is pickled it is packed in salt or in a concentrated brine. The salt inhibits harmful bacteria. In very high concentrations the meat will be "cured" and be dry. In lower concentrations lactobacilli will still grow. The bacteria produce lactic acid which finishes the preserving of the meat.

Neither method requires refrigeration, of course, but modern home recipes generally call for refrigeration anyway to remove risk. Modern recipes sometimes also reduce the salt to a more gentle level which then does require refrigeration. Corned beef and ham are the canonical salt-cured meat recipes in the US.

It is rare to see home recipes that allow meat to ferment. I think we've lost the taste for it in most contexts. Fermenting meat is still often done commercially where the variables can be controlled better. Pepperoni and various Norwegian fish products are examples.

As a rule of thumb, only use food preservation recipes from trusted, modern sources. It is not uncommon to see antique recipes that are unnecessarily risky.

If you are interested in the history, chemistry, and politics of salt-cured meat, the book "Salt" by Mark Kurlansky is amazing.

As Stephie noted in her comment, you were right to throw away your meat. Your recipe did not seem to use enough salt to ensure prevention of harmful bacteria growth. Pickling brines generally call for enough salt that an egg floats.