Is an accidental murder a punishable offense?

Depends if it was a stupid enough accident. Different jurisdictions might define “stupid enough” slightly differently, but the general idea is probably the “reckless disregard” standard. In other words, if you were intentionally doing dangerous acts without an intent to kill, but with reckless disregard for others’ safety, then you can be held criminally responsible for manslaughter.
For example, suppose you live in an apartment building. One day, you get a new pistol. You become curious how loud the pistol is when fired, so you fire a few shots through a wall. The bullets go through the wall and kill your neighbor. You had no intention to kill your neighbor, but you’re a big enough idiot to not recognize the risks involved in shooting a pistol under these circumstances. Some states have separate laws to handle specific types of idiocy. For example, many states have vehicular manslaughter laws to cover the specific case of homicide while driving drunk.