First, do you have a written lease? If so, does the lease state whether the landlord is responsible for said repairs? If so, send the landlord a letter requiring him to abide by the terms of the lease, if such repairs are his responsibility.
Next, under what is known as "the implied warranty of habitability," a landlord has the obligation to ensure that a dwelling is free from hazards that affect the health and safety of his tenants. Clearly, a sewage backup can pose a health risk for the tenants. In some states, you can withhold rent if a landlord fails to make certain repairs; in other states, you can file a rent escrow action. The important thing is to send your landlord a written request to fix the repairs and allow him a reasonable time to do so. If he fails to fix the backup, then you may have options. In Michigan, you can file a rent escrow action and/or deduct the cost of repairs from your rent. A helpful guide can be found at the following link (note pages 26-27):
Consult a local landlord/tenant attorney for more guidance.
DISCLAIMER: Brandy A. Peeples is licensed to practice law in the State of Maryland. This answer is being provided for informational purposes only and the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship. For legal advice relating to your specific situation, I strongly urge you to consult with an attorney in your area. NO COMMUNICATIONS WITH ME ARE TO BE CONSTRUED AS ARISING FROM AN ATTORNEY-CLIENT RELATIONSHIP AND NO ATTORNEY-CLIENT RELATIONSHIP WILL BE ESTABLISHED WITH ME UNLESS I HAVE EXPRESSLY AGREED TO UNDERTAKE YOUR REPRESENTATION, WHICH INCLUDES THE EXECUTION OF A WRITTEN AGREEMENT OF RETAINER.
Under what's called the doctrine of constructive eviction, you may be able to terminate the lease and vacate the premises. This doctrine allows a tenant to leave without liability if a defect in the premises that is not caused by the tenant renders it uninhabitable for the purpose for which it was intended - in this case, as a residence. Needless to say, the smell of human waste can be intolerable, especially when it gets into the drywall and carpet. It also breeds bacteria and diseases, which creates a health hazard, especially if it's backing up into your bath tub. You also need to be sure it's not backing up into your washing machine, drinking water, etc. (I'm not a plumber, so I don't know if that's even possible). A chronic health hazard would be good cause for constructive eviction.
You say that the landlord won't FIX the backup mess. You make it sound like it's a recurring problem. If so, constructive eviction may be your best (and most immediate) remedy. CONSULT AN ATTORNEY BEFORE EXERCISING THIS REMEDY!!! Constructive eviction is tricky and there is a process you have to follow before using it.
If this is a cleanup problem, you may still have a case for constructive eviction depending on how bad the problem is. You may be entitled to prorate your rent for partial constructive eviction if there are areas of the apartment you can't use due to the flooding damage.
Otherwise, I agree with Ms. Peeples, but I would add that if the landlord fails to fix the problem in a reasonable time after you give him notice, the landlord has violated the warranty of habitability and breached the lease. This would allow you to terminate the lease and move out due to his breach.
Again, consult an attorney in your area.