I think that the solar system is a very small part of the visible universe-- The farthest planets from the sun are at about 40 times the earth-sun distance, making the planetary system about 80AU or 8 billion miles across.

We think we know how far it is to the nearest star from the drift it makes in the sky as the earth goes around the sun. This drift (called parallax) is about a second (1.3) of arc (1/3600 of a degree) when the earth moves 180 million miles. So if light travels in nearly straight lines, the nearest star is on the scale of 1.3*57*3600*1 AU = 268,000 AU = 25 trillion miles.

Scientists have studied stars for about 150 years, and have developed models which seem to correspond to observations and using these models we can predict the brightness of a star from its spectrum. For stars in the Milky Way, this gives a size for the galaxy of 90-100,000 parsec = 90,000*206,000 AU =1.7x10¹⁸ miles.

Finally, galaxies seem to bear out a distance scale by their red shift (again measured spectroscopically) to indicate a visible universe of radius 14 billion light years. (A parsec is 3.26 light years). So that makes a visible universe with a radius of
9x10¹⁴ or 900 trillion AU = 8x10²² miles.

Is it 'really' this big? It is hard to say... the current understanding of the physics involved (assuming physics act there as it does here) seems to indicate it. Indeed, most modern theories of cosmology predict a 'real' universe (beyond what we can see) that is millions of times larger yet.

I'd be very surprised if the solar system distances were off by 0.0001%, surprised if the nearest start distance were off by more than 2%, not surprised if the galaxy size was off by 30% and/or the universe size off by 50%. (The age estimates and hence size of the visible universe have changed by 40% in the last 20 years...and are destined to change again as theories improve).

As to the technology -- there is no telling. 20 years ago, high energy physics needed kilometers and Megawatts to make electrons at 1 GeV --nowadays, it can be done in a meter with a 10 Watt laser.

Beyond our solar system are other stars. We know that they exist because we can see them.

So far as we understand, the universe is infinite -there is no "edge". Now, we can only see about fourteen billion light years out, but that is because fourteen billion years is all of the time that light has had to travel, not because that is where the edge of the universe is. There does not seem to be any direction, if you will, to the shape of the universe; matter is certainly clumped together into clusters of galaxies, but there doesn't seem to be a center in any direction, so we assume that it is probably infinite.

As for technology, it is not a question of technology. The limit of our sight is imposed by the finite speed of light and the universe's finite age. Every second that passes, we can see one light-second (about 300,000 kilometers or 186,000 miles) farther, because light has had that much more time to travel to us.