Assuming planet Earth is doomed three hundred years in the future. Predicting the future is never easy. There could be new physical discoveries that could be implemented as practical technologies. Any one of which could make sending people twenty thousand years in the future with ease, or thirteen million years even. What can be expected is the development of quantum computing which will open up a wide range of scientific understandings.
The two obvious options are biosuspension and relativistic time dilation. Other answers have dealt with these options already. Of the two, biosuspension is the more probable and the proposals have some ingenuious suggestions. While achieving relativistic time dilation space travel is far less likely within a few centuries.
This answer will propose exotic solutions based on established physics and technologies, although this does assume advanced extrapolations of these solutions.
Stasis fields based on atom traps
Atom traps are a technology used in the laboratory mainly for experimental work on quantum mechanical systems. Essentially single atoms are held in place by confining photons. If this technology could be extended to trapping multiple atoms in complex systems and especially if this could be scaled up to the level that all the atoms in objects like people and equipment.
This would be effectively a stasis field and if these stasis fields can be maintained for many thousands or even millions of years, then a selected group of people can be sent into the future.
Gravitational time machines
The method uses the other form of time dilation. Namely, gravitational time dilation. To make this technique work we will make three assumptions about how this could be feasible.
One, gravitation is based on gravitons. Two, gravitons can be collected and, three, can be confined to specific locations for sufficiently long periods of time.
This gravitational time machine operates in this way. The time-shifted group are placed inside the time-machine. Gravitons are collected and confined to the volume of the machine. As the internal gravitational field inside this chamber rises this will increase the amount of gravitational time dilation. If the density of gravitons rises progressively over the time these humans who are inside this machine will experience less and less time passing. This will enable them to travel into the far future.
Where will the time machines be kept?
If the Earth is doomed, then either time machines, the atom trap stasis fields and the gravitational time dilation system, might be destroyed along with planet Earth. Therefore, they need to be kept somewhere safe.
By the 24th century, space technology should be developed enough to launch one or either of these time machines into interstellar space at a low velocity of around 300 km/s or one-thousandth of the speed of light. Travelling at this velocity a time machine would travel ten light years over ten thousand years. At this point the time machine/spaceship can decelerate and head back to the solar system arriving a further ten thousand years later. This is the minimum time suggested for recolonizing the Earth that has recovered from its doom.
Obviously if longer time periods are needed to be traversed, then longer distances can be travelled at a suitable low velocity.
Admittedly these time travel technologies aren't the time machines as commonly found in science-fiction, they are based on known physical principles and existing technologies. Although they have been extrapolated to accommodation putative advances over the next several centuries.
