Once it becomes cheap enough then religious or similar cults will do it, and the cost of doing isn't a great deal more than colonising the solar system where are are economic benefits.

That assumes reaching 5 light years in say 50 years

However once it's done a few times, then it can't be stopped. If you're going at say 1/1000th of the speed of light (50 years to travel, 450 years to settle and build the next wave is 1% the speed of light) then your successor species will have colonised the galaxy in 100 million years, and we know that hasn't happened.