Let's imagine that people, if they'd want could freely leave Earth to pursue their own destiny on other planets etc. How far could we get assuming following:
Assumption No. 1: It's trivial to get any item up to 260 metric tons ( fully loaded Boeing 777F weight ) into space ( 800 km from Earth's surface ). Anyone could save up money to buy the Space Elevator ride and get whatever they'd want into space.
Assumption No. 2: Every person has a life-span is ~500 years thanks to curing cancer and discovering how cells can replicate almost indefinitely. Then, however, the brain-structure starts to fail rapidly and no cure is found.
Assumption No. 3: People can be put into stasis up to 100 years and can reenter stasis after a year.
Assumption No. 4: No significant progress has been made in any field from today's standard. Can't use dark-matter as fuel, no opposite pole of gravity is discovered etc.
Assumption No. 5: Our hero is an engineer, has good technical education, young - only 30 years old and totally sick of Earth and how things are. He's not rich, but is definitely high-end of middle class.
So our hero wishes to take to space and establish a utopia in his understanding (doesn't matter what it is). How realistic is it for him to reach an Earth-like planet in his lifetime?
There's no pre-defined blueprint of a spaceship, whatever he puts together he'd fly, but for the sake of simplicity let's say that base components are available and wouldn't cost too much just like getting a cell-phone was a luxury some 25 years ago and now everyone has a smartphone!
Things that our hero decides to take with him:
Worlds library - the largest collection of books in digital format on every subject in most of the languages. Not complete collection of every book ever, but the largest that we have at this point.
A super computer - a PC that would seem like a super computer to us, some x10 000 times more powerful than average computer today.
A telescope - powerful enough to see a persons face on The Moon from Earth if the sky is clear.
Large solar panels that would be enough to power a large apartment if there's light.
A 3-d printer that can print both circuits, plastic, metal and compose diamond. It can actually refill its matter storage from rough material (eg. CO2 → diamond)
So, knowing that space is full of radiation, small razor-sharp particles, magnetic waves etc. How possible is it to construct a space-craft that would be sustainable enough for at least 120 years in-between repairs? And how safe would it be?
And finally - how far could the hero get in his lifetime if the speed he could travel is roughly twice the speed of sound (relative to Earth's atmosphere) while in vacuum of space and around 900 Km/h in atmosphere similar to that of Earth's density?
EDIT: if some of you are unimpressed by the stats that the said spaceship would have — please note it's built by finances of a single individual. It's very very possible that once landing on a planet the spaceship would not be able to leave the orbit again. So when answering the question — think in terms oh what could go wrong on his way to Planet Paradise. Also feel free to specify what our hero should prepare for/bring with him to even attempt a journey like that.
EDIT2: Made some calculations and found out that even with speed like 40 000 km/h it would take thousands of years to reach a different solar system ( and no promise that it could have Earth-like planets ). So unless there's a way to travel at least 1/4 speed of light it would seem impossible to get very far and be able to see different planets.