The answer is, usually, no. Some very rare birth centers may offer epidural anesthesia and have obstetricians and anesthesiologists on staff. Birth centers are normally staffed by midwives (either Certified Nurse Midwives or Certified Professional Midwives) and don't have the ability to offer epidurals. Besides, it might also be against these midwives' ideology to offer this type of pain medication. There are several possibilities open to you if you would like a birth center delivery, or if birth centers appeal to you though:Birth centers that are attached to hospitals and those who are on hospital grounds may have easy transfer procedures. If you decide half-way through your birth that you don't want to go on without an epidural, you might be able to get to the main maternity ward in a very short amount of time. You may like to try non-epidural types of pain relief. These include IV meds, laboring in a birth tub (water is said to reduce labor pain), moving around during contractions, and gas and air. Hypnobabies is another option that works well for some women. If you want midwifery care and an epidural, this could be available to you in a hospital. You might still get the kind of personal care you wish for, with all the appropriate technology and professionals being right there.
Natural birth, mostly meaning unmedicated birth, is what women and midwives who work at a birth center are normally dedicated to. Still, there are many reasons to want to choose a birth center for your delivery that are not related to not wanting pain meds one on one care, and a more luxurious, less medical environment, for instance. So if you want those things, but would also like epidural anesthesia (or at least want to have that option), can you still give birth at a birth center?