Fun idea. Probably have to think about it.

First, save Earth or not save Earth. If the problem is to save mammals, then saving Earth is probably necessary.

Well forgo the question as to whether, given the size of the Universe, Man is that essential to anything besides Man. Realistically, what would be lost if Man disappeared now? Would any other part of the Universe notice or be affected, now or in the future?

Let's say Earth is gone. Terraforming another planet? Might be tough. There's an elegant adaptation over billions of years by life to Earth's special and myriad conditions. Could a cheap, efficient and long-term way be found to create enough of those conditions on another local planet to allow more evolution? Couldn't count on it.

Out of the solar system. Probably couldn't manage to keep entire reproducing examples of species going long enough to find another "home".

I would think, knowing what we know now, that the only realistic way to keep life going would be to broadcast some elemental, self-replicating form of DNA into the void. Would have to give up on entire mammals, entire critters of any sort, really.

Elemental so as to give it the best chance of surviving the trip in a form which would allow it to thrive.

May want to figure a way to help it survive the various forms of radiation it may meet along the way.

At first might want to direct the stream toward where likely solar systems are going to be about the time it gets there.

If the local environment were hospitable, would a DNA life have formed, anyway? Would our "messenger DNA" turn out to be needed, or have a niche, or even be a nuisance? Dunno.

And, I wonder if the entire mass of Earth, if converted to DNA (which clearly isn't possible), would be enough to give any of it a chance to survive before the Universe turns very cold or very hot.

I don't think we can "save" Man, or mammals, may be able to give life a head start on places it probably would inhabit anyway.

Well, if one were determined to "save" life as it is found on Earth, knowing what is known now, it would look like a daunting project.

And, my suspicion, given the size of the Universe, is that life is probably pretty common, but the complexities of it makes it a local affair wherever it is.

Arf