Getting ill in a foreign country is nothing to take lightly -- and medical expenses in Japan can be quite costly as well. If you live in Japan you can choose to enroll in Japan's National Health Insurance. Under this plan you pay a monthly premium, as well as a 30% deductible of your medical/dental bills immediately on treatment; the government pays the other 70%.
You can sign up for it at your local city ward or town office, and aside from that there is never any paperwork or red tape to go through. The ward or town office determines a monthly premium to pay based on your previous year's income. If you're new to the plan the amount can be quite low -- around 1350-4500 yen per month. However, every April at the next premium adjustment you may find the amount escalating sharply, even if your income changes very little. A first year of 4500 yen a month, a second year of 13,500 yen a month, and a third year of 35,000 yen a month is typical (Japan has an exploding number of old-timers). Cosmetic/elective surgery, child birth, and abortions are not covered, however.
Previously, some ex-pats who planned to leave Japan and tried to cancel their government insurance were given the standard response to Japanese, "You can't cancel it." However, the truth is you can, and if someone lays that response on you to may have to reiterate that you're leaving and push a little more for them to terminate the insurance. They may ask to see your air ticket to prove you're actually leaving. So if you plan to change from the government plan to a different plan while in Japan you may have to do it before you take a vacation abroad somewhere. If you're definitely staying in Japan long-term, and your employer won't pay the insurance rate for you, then you might consider private insurance either with Japanese or foreign companies.