Is it possible to prevent pregnancy without using condoms or pills?

Yes. Abstinence (not having sex) is very effective but not much fun. There are many other methods of birth control. The diaphragm doesn't involve hormones but requires use with each episode of sex. There are the patch, the ring (more convenient delivery systems for the pill); the depo-provera shot, the Implanon implant (long-term hormonal bc without estrogen); and iuds (put into uterus lasting 5-10 y.
Other options. The most effective methods of birth control are the so-called long-acting reversible contraceptives (larc) methods which are extremely effective( as effective as sterilization), long acting 3-10 years and include a rod that is placed beneath the skin of the upper arm, or two types of iuds.There are also other options-- the patch, a vaginal ring, or other non hormonal options that fail more often.