There's a definite problem with interstellar long term space ship travel. First of all it would take lots of resources and thus money, the amount of metal and other nonrenewable resources needed to make it big enough would be incredible.

First of all, you'd need to create a artificial ecosystem involving quick growing plants that have a high photosynthesis to volume ratio, As well as a system to keep them reproducing and growing, as well as plants that would be used for food, materials and medicine.

You would need a bacteria and decomposer system going to naturally break down toxins and pollutants and contaminates in the water, so sewage could be be filtered and the water reused, as well as for trash, basically a compost pile with different stages, so all matter is recycled.

You'd need some sort of renewable energy source like something working off the spaceships own gravity, plus solar panels.

You would need mini factories, for the production of medicine, food, equipment, clothes, all products needed.

The ship would need Hospital bays, methods of transport, and lots of places for the huge population needed to keep everything running to live.

Of course a great deal would be done by the technology, all automated, but you would need people to repair and operate and monitor some of it at least.

This big hulking thing would need multiple huge engines and systems and defenses and thrusters, which would require lots of power to get it going and maneuvering, all of which would need to be run by electricity. All of this and other computers and servers and parts would take up considerable amounts of room.

All of this, would require world cooperation, sharing of resources, extensive mining and extracting, destroying our earth to make a new home, We would need to do all this, for hundreds and hundreds of years of work, while still having our economies and governments... Extremely unlikely.

Now here comes the worse part...

The space ship would need to recycle all it's matter, so when parts break off or are destroyed in space, that matter is lost. In order to stay sustainable it would need asteroid and other space rocks mining systems, and it's not sure if youd be able to extract the matter you need, or if you even have discovered this element before. You would need the machinery and factories to make parts to repair and make things.

Why all this sustainability and recylcing you ask? In interstellar travel you cant have a transport bringing materials from the home planet to the ship, at least not while the ship is far away or moving away. You can't just stock pile everything you need, because at the distance and speed you would be going it would take millennia and millennia to reach the destination, generation after generation of humans, meaning you would need a darwinistic nazi society to kill off any genetically inferior humans to ensure that bad genes did not contaminate the gene pool and make the population all susceptible to those genetic illnesses. Same with all organisms on the ship, to prevent them from being all susceptible.

All this would be required for us to explore the galaxy, all this for interstellar flight. A single thing could end it all. In the end, it is highly unlikely it would ever happen. The problem with the speed of light is, how do you know if what you are heading for is still there by the time you reach it? Going faster than the speed of light would have serious implications if it was actually possible...

Sorry to ruin those sci-fi dreams...