Elon Musk to the Rescue
If you asked this same question 10 years ago, I would be as pessimistic as many of the other answers. But this is 2020. The Falcon Heavy is in production. Starship is on the way. Elon Musk has, many times, said that his goal is to establish an outpost on Mars to save humanity.
One "freebie" technology? That would have to be suspended animation (and I don't mean canceling the Flintstones). It is not as far away as you might think. I don't think it could make us live forever, but if we can sleep through the trip and wake up when we get there then we can save a lot of payload otherwise needed for food, water, medicine, etc. If this is a 100-year journey (using Project Longshot as a starting point) and we have sleeper technology that gives us 100:1 metabolic processing then the human bodies will have effectively aged only 1 year (so they are still "young" when they get to the destination) and the nutrients consumed would be equivalent to 1 year - not far off from what Starship would need to take for a Mars trip. While sleepers would need extra equipment to keep them fed, healthy & safe, they would need essentially crowded bunk space but no common areas, exercise rooms, etc. and all the extra payload space & weight could be devoted to equipment needed on arrival. Unlike one book I read (can't remember the name at the moment), we need to make sure everyone sleeps through - if one person wakes up 1/2-way through and can't go back to sleep then he would eat (in 50 years) 1/2 the food intended to get all 100 people through the first year on the new planet.
How many can we save?
We can't save everyone. But we can easily save thousands. The key is using Starship to get things in orbit and assembling, in a minimalist fashion, in orbit. The main Starship is designed to be fast-reusable, which is a key. Something like:
- 19 Starship flights for a 3-level hexagonal ship - 1 in center, 6 in 2nd level, 12 in 3rd level. Inner 2 levels have 200 sleepers each = 1,400 people. Outer level filled with food for the first year at the destination and equipment to set up a small colony - water purification, seeds, medicines & medical equipment, computers, tools, solar panels, Teslas, etc. Outer level also provides radiation shielding for the inner levels.
- 8 Starship flights for a propulsion module. 1 in center to separate this from the main section (not sure this is needed but it is a common style of ship in many sci-fi novels), 7 in 2-level hexagon - center for propulsion (ion? fusion?) and 6 for fuel (reaction mass, not ordinary rocket fuel)
That is a total of 27 Starship flights per 1,400 people. SpaceX, with some BIG government help (remember, so far SpaceX has done the bulk of its development using some government funding but primarily funded by paid commercial, government and military flights and private investment) gets the Starship assembly line running at [Liberty Ship](https://en.wikipedia.org/wiki/Liberty_ship speed then we get Starship serial production within 2 years, with ship (sleeper, cargo, propulsion) design running simultaneously. Assuming Starships can get a weekly refurbish rate and 100 (= 2 years) uses per Starship before retirement (while major overhaul might make sense, I'll assume some occasional damage-beyond-practical repair, plus the production line for new Starships will make major overhaul less cost-effective), that would mean a total fleet of ~ 60 Starships but won't have full launch cadence until ~ 30 have been produced. So early production will be key. On the other hand, the larger ship components will be needed at a steady 2 per day throughout the launch window. Start with the equipment & fuel modules. Sleeper modules would go up last to give those people maximum time on Earth - last 2 years would be almost all sleepers.
With 4 launch pads (2 Florida, 1 Texas, 1 California) running on an alternate-day basis (try for daily but there will always be some days you can't launch due to weather, delays in payload processing, etc.) that would be (2 per day * 365 days * 8 years) / 27 per ship * 1,400 people per ship = 302,814 people. Figure 10% loss due to ships not completing the journey due to mechanical failures, 10% loss of sleepers due to problems "waking up" and you've still got 245,279 - a quarter million people is a nice large genetic pool to restart humanity.