This is amplification on previous answers citing Project Orion (aka Nuclear Pulse Propulsion). Read the provided reference for history and technical background.
Background
Research & testing performed from 1950s - 1970s indicated that using that level of technology we could built a 8,000,000 tonne craft capable of achieving orbit and providing a bit of extra $ \Delta V$, perhaps enough for $V_{esc}$. However, only about 1/3 of this mass is payload mass (2,700,000 tonne).
The problem
Let's assume we need to lift every human being off the Earth and there's zero population growth:
Current population ~ 7,000,000,000 each
Average mass ~ 100 kg / each
Total mass ~ 700,000,000,000 kg
Total tonnes ~ 700,000,000 tonnes
Assume that we need 20x this mass for equipment & consumables for keeping people alive in transit, colony construction equipment, plus sundry other items. Also assume that the ships are constructed in a modular prefab fashion that allows us to directly use them at the destination or dissemble them and use the parts in existing colonies:
Total lift requirement in tonnes ~ 14,000,000,000 tonnes
Per craft payload mass in tonnes ~ 2,700,000 tonnes / craft
Total required craft ~ 5200 craft
Estimating timing
Wikipedia states that world production of container ships was ~11,000,000 tons in 2011 and our Super Orions would require similar (but more stringent) levels of construction difficulties (remember we can use normal construction materials like steel). So without straining we could build on average 1 of these Super Orions per year but each would require multi-year construction (say 5 years like US aircraft carriers or perhaps even 10-15 years). We'd be done in 5,200 years.
With straining, (I would guess) we could build 10x this number. We'd be done in 520 years.
With all out desperation, the upper bounds of what we could make would be determined by our critical resource production (such as steel). Assume our 8,000,000 tonne craft are composed entirely of steel and that this is our limiting resource. This site indicates world steel production is around:
World steel production ~ 150,000,000 tonnes / year
Max Super Orion production ~ 20 ships / year
We'd finish making our Super Orion fleet in 260 years.
Estimating cost
Let's assume that the amount of labor and difficulty of constructing these Super Orions equates on a tonne per tonne basis with the expense of building nuclear aircraft carriers.
US nuclear powered aircraft carriers mass about 100,000 tonnes of displacement and cost 26 billion (USD).
Each Super Orion will cost ~2.1 trillion (USD) or about 2/3 of the 2014 US federal expenditures. The fleet will cost 11,000 trillion (USD).
Destinations
The Earth is the garden spot of our Solar System. Very few places have both the readily available volatiles, metals, and other materials (soil?) that we'll need to survive. IMO, Mars and perhaps a few of the main body asteroids (Ceres is looking really good right now) might fulfill this role.
But really only Mars has enough volatiles, metals, and room to host a significant portion of the human population. Where are we going to put everyone?