The system you're describing here may be a contact binary, where the two stars have come close enough to actually touch. These systems may actually be stable for millions (or even billions, in extreme cases) of years; the lifetime of the system as a whole - outside the contact binary period - will certainly be similar to the normal lifetimes of the individual stars. The components may be massive O-type stars (see VFTS 352), or even low-mass M dwarfs (see e.g. Qian et al. (2015)). In other words, they run the gamut on spectral types.
The progenitor stars may be variable stars - not great for life, but maybe not that terrible. They'd likely be W Ursae Majoris variables, if the stars are of spectral types A through K, with periods of perhaps 0.25 to 1.0 days. The variations are not drastic, spanning less than one magnitude (i.e. less than a factor of 10). It's not clear whether or not life could adapt to such circumstances, but it's good that the variations are so short.
An interesting case to look at is KIC 9832227, a pair of stars not wholly unlike the Sun (one is a G-type star). The combined luminosity is about 3 solar luminosities.
Interestingly, the stars are predicted to merge in about five years, in 2022. They are expected to produce a luminous red nova. (A new paper has shown that the merger prediction was incorrect and a result of human error - no such merger will occur in 2022). The system may one day produce a luminous red nova, which is what likely happened in the case of V1309 Scorpii, in 2008 (see e.g. Tylenda et al. (2011)). The system increased in luminosity by about 3.5 magnitudes for about six months before the main outburst, where it brightened by an additional 6.5 magnitudes - not great for life. It then subsided over the course of several years.
I'm actually pretty confident that life could survive in this sort of a system during the main period of evolution. Changes in stellar activity shouldn't be too bad; variations of less than one magnitude over short time scales are almost certainly not problematic. However, the nova itself would likely be very bad - very bad indeed.
Let's again look at V1309 Scorpii. In the period of slowly increasing brightness, the stars lost mass, possibly forming an excretion disk (the opposite of an accretion disk, in a sense). It is unknown how this disk will behave; it may dampen the star's luminosity by some amount. At any rate, the luminosity will be closer to the original value after some time; the system has already dropped in brightness considerably.
It's possible that planets could form from the disk. This may be how some (though not all) pulsar planets form, and it could likely happen around the results of luminous red novas. Life has a fighting chance of starting anew. The long-term behavior and evolution of these systems is not well-known, so it's currently not possible - as far as I know - to rule this out.