Halachically is my landlord allowed to evict me if I have nowhere to go and I can’t find an apartment for a while?

According to the letter of the law, if the contract ended, he is allowed to evict you from the apartment. Beyond the letter of the law, he can allow you to stay in the apartment after the contract ended for another period of time. For a practical question, it is recommended to contact the local Rabbi or Rabbinical judge.