It's highly dependent on how fast we develop certain technologies, and can expand into space. We theoretically have the capability to build spacecraft that can last generations today (the voyager probes), and they weren't even meant to last that long, so if we did want to build a spacecraft that can tough it out, it could last a long while before breaking down.

One big problem is propulsion, in order to get to these stars within a century or so requires us to go at least 5% of the speed of light, which is no small feat. Thankfully, many scientists have devised ways it could be done, but these are speculative concepts, and aren't comparable to blueprints since nothing has quite left our imaginations (yet).

Some of the propulsion systems that are capable of this, in no real order:

Orion pulse propulsion - possibly feasible with not just current technology, but technology from the 60s. Instead of the hard problem we have with sustainably containing fusion in a compact way, why don't we set off specially designed nuclear bombs outside the ship, and use its momentum to move forward? You would need a lot of nuclear charges to get up to speed, but a lot of people back in the day worked out that it was possible. It's just a politcal dead end due to the amount of weapons grade fissile material you need, and the nuclear test ban treaty.

Then there's fission fragment - also possibly feasabile with current technology. It uses rapidly spinning discs coated in fissile fuel (spinning to prevent melting from the intense temperatures), and as the fuel undergoes fission, the 'fragments' get expelled with a magnetic nozzle at incredibly high velocities (up to 3% of the speed of light) which is pretty great and helps give a high specific impulse and overall top speed. I don't know how fast you could ultimately go and still slow back down on the other end, but I assume you'd be able to reach Proxima Centuari within a couple hundred years or so.

An indirect form of propulsion are laser sails. It uses the same principle as a light sail, using the momentum of photons to accelerate a reflective sail with a smaller spacecraft. But where sunlight drops off with distance, and gives a limited top speed (something up to a few hundred km/s. Good for interplanetary - even though you wouldn't actually max out, but not enough for interstellar), a laser can concentrate that light to such a degree that it will be able to accelerate you to up to 10-20% of the speed of light (~40-20 years to Proxima Centuari), and can have multiple relay stations to take over acceleration as the lasers will also eventually become diffuse. These can be powered by solar arrays closer to the inner solar system, and nuclear reactors further out. The only downside? You need these same stations on the other end to slow back down, or a secondary propulsion system like the ones above. But, this does save fuel overall, because instead of bringing all that fuel to accelerate and deccelerate (which requires an incredibly high mass), you only need enough to deccelerate (which is much lower).

These are all within current technological bounds, and goes to show that even if nuclear fusion never pans out in a practical way, we still have a means of accomplishing interstellar travel in a more reasonable period of time, not the tens of thousands of years commonly brought up (and only applies when talking about chemical rockets - which we'd never use if we were actually serious about an interstellar mission to another star).

Actually building and launching one of these ships is outside our capability because even if we have most or even all of the technology, we'd still need a ship spanning hundreds to meters to a few kilometers, massing hundreds of thousands to millions of tonnes, with overall power outputs exceeding the terawatts. And for that we don't currently have any industrial or manufacturing capability in space that can do that. I don't see humanity being ready to send out an interstellar probe for maybe a century, maybe up to two centuries. Depending on how practical it is to scale up space outposts, ISRU, and the like in the next few decades.

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On the bright side, we might not have to wait quite that long to get confirmation of alien life. JWST will be sensitive enough to pick out likely candidates by examining their atmospheres for things like carbon dioxide, water vapor, oxygen, and methane (really looking forward to the TRAPPIST-1 findings in this case), but probably not enough to confirm life outright. However, it will definitely let astronomers know where to look in the future. The Nancy Grace telescope will be able to directly image Jupiter sized exoplanets in somewhat better detail than now, and be a testbed for a coronagraph that can image potentially habitable Earth sized planets. PLATO and ARIEL (to be launched in 2026/29 respectively) are ESA missions to look for exoplanets, particularly Earth sized habitable zone planets, while ARIEL will follow up on known planets and will be able to also characterize their atmospheres, though I don't know how much more/less sensitive their instruments are compared to JWST.