The biggest challenges are Time and Energy ! It will take a lot of onboard energy that has to run for centuries !
Total non returnable trip time to nearest solar system (4 light years ) in proposed ship at max speed 4% speed of light , 12,000 km/per/sec. 300 years.
If we built a sufficiently large space ship in earth orbit maybe with internal volume of at least 10,000,000 cubic meters (approx 3 times the size of NASA Vehicle Assembly Building ) and would be constructed of very light composites and super alloys to keep weight down . It will also be fitted with 5 nuclear power stations that can run for 50 years without refuelling (fast breeder reactor type) and 350 years worth of fuel and spare parts totaling 30,000 tons . It will also have a 3D printing workshop where anything can be duplicated including high strength metal components.
Accommodation for 2000 crew and families , plus food for 10 years and a highly efficient recycling system and very large hydroponic growing system . Plus 100,000 tons of water . There would be no shortage of heat as the power plants will put out prodigous quantities of waste heat that will be utilized throughout the ship which will be transported by water .
Everybody on board would work and families would be expected to start having children in a designated time period to replace people that had passed on , pop. limit 2500–3000 people . The ship will have a very large diameter and will spin at approx 1.2 rpm producing normal gravity in the Toroid (main living) and various other gravity gradients through out the rest of the ship . Large area on center line of ship reserved for very low gravity sports .
Nothing could be discarded from the ship in interstellar space , everything will have to be recycled including people that pass on, there will be no such thing as garbage ! Schools/ Universities on board plus a plethora of entertainment possibilities for everyone . Everything that is required to keep people sane for centuries.
Propulsion ? With a large amount of electricity available (40,000 megawatts ) a large number of NASA Evolutionary Xenon Thrusters (NEXT) would be the most viable propulsion engines today using xenon gas as fuel . The ship would need 300,000 tons of Xenon Liquid (102,000 Cubic Meters ) and a total ship weight of 600,000 tons.
Say 35,000 megawatts available to engines would deliver 121,730 kilograms of thrust from the Xenon fueled engines .
It would take 100 years of acceleration to achieve 4% of c and travel just over 1 light year then cruise for 50 years to traverse 2 light years then 100 years of de-acceleration as it approached new star system . Total trip time 300 years , total distance traveled 4 light years to Alpha Centauri Star system .
At this velocity relativistic effects are rather minor. Ship time will be dilated down to 3.9967987189749747 years instead of 4. So the occupants of the ship will be approx 4 minutes younger than those left on earth !