What are my options for trying to leave a rental unit with no termination clause in the contract?

You have no right to terminate the lease early, unless--

1) The lease itself gives you  that right (e.g. allows you to terminate on some notice)

2) The premises becomes uninhabitable (unsafe; or legally uninhabitable, such as the city orders that the landlord may not legally rent at that building)

Other than the above, by signing the lease, you (and anyone else on the lease) is legally obligated to pay the full rent for the full term. If that is not paid, the landlord may sue you for the remaining balance (e.g. if you move out 5 months early, for 5 months of rent). If the landlord sues you and wins--which he almost certainly would, from what you write--he will have a judgment against you that he may to enforce in various ways for several years (e.g., he can wait until you are employed and then seek to garnish wages).

Your roommate, if he or she signed the lease also, is also liable. The landlord could elect to sue either one or both of you. If you are sued, you could sue your former roommate for his or her contribution.

If the lease does not prevent you from assigning or subletting it, one option you have is to look for someone to take it over from you or sublease from you. Even if you have to take something of a loss  (e.g. rent it for less than you are paying) to get someone to take it, this would let you minimize your losses.

Note that if you don't pay rent and move out or are evicted, the landlord could also take unpaid rent out of your security deposit.