Would this barrier affect the Solar sysem's heliopshere?
The heliosphere is the magnetosphere, astrosphere and outermost atmospheric layer of the Sun. It takes shape in form of a vast, bubble-like region of space. In plasma physics terms, it is the cavity formed by the Sun in the surrounding interstellar medium. The "bubble" of the heliosphere is continuously "inflated" by plasma originating from the Sun, known as the solar wind. Outside the heliosphere, this solar plasma gives way to the interstellar plasma permeating the Milky Way. As part of the interplanetary magnetic field, the heliosphere shields the Solar System from a significant amount of cosmic rays, including hazardous ionizing radiation (e.g. gamma rays). Its name was likely coined by Alexander J. Dessler, who is credited with first use of the word in scientific literature in 1967.1 The scientific study of the heliosphere is heliophysics, which includes space weather and space climate.
Flowing unimpeded through the Solar System for billions of kilometres, the solar wind extends far beyond even the region of Pluto, until it encounters the "termination shock", where its motion slows abruptly due to the outside pressure of the interstellar medium. The "heliosheath", is a broad transitional region between the termination shock and the heliosphere's outmost edge, the "heliopause". The overall shape of the heliosphere resembles that of a comet; being roughly spherical on one side, with a long trailing tail opposite, known as "heliotail".
Two Voyager program spacecraft explored the outer reaches of the heliosphere, passing through the termination shock and the heliosheath. Voyager 1 encountered the heliopause on 25 August 2012, when the spacecraft measured a forty-fold sudden increase in plasma density.2 Voyager 2 traversed the heliopause on 5 November 2018. Because the heliopause marks the boundary between matter originating from the Sun and matter originating from the rest of the galaxy, spacecraft that depart the heliosphere (such as the two Voyagers) are in interstellar space.
Voyager 1 crossed the heliopause at a distance of 121 AU, so the heliosphere has tiny dimesions compared to the barrier at a radius of 1/6 LY or about 10,000 AU.
If matter and energy enter the barrier freely, the heliosphere should behave much like it does in real life.
If matter cannot leave through the barrier, it would accumulate inside over 550 million years as Alexander's answer says.
It is possible the density of gas and dust inside the solar system would become much higher, and affect the orbits of the interplanetary dust inthe solar system, perhaps producing detectable results.
And it would tend to accumulate on the back side of barrier. Thus it might have formed a new solar system dragged along by the barrier and about 1/6 LY or about 10,000 AU behind our solar system.
Alexandaer calculated that the accumulated mass would be over two times the mass of the solar system. If a star did form in the trailing region, most of the later incoming matter would eventually fall into that new star and increase its mass.
A star with about 2 times the mass of the Sun would be similar to a spectral class A4V star with 2.03 times the mass of the Sun, and at a distance of 1/6 LY or 10,000 AU it would be the brightest object in the sky of Earth except for the Sun and the Moon.
Incoming matter that entered through the barrier and then hit the back side of the barrier would be prevented from exiting, and thus its velocity would be changed. That would impart some of its velocity to the barrier, which should transfer the velocity to the barrier generators, which should be tied to the Solar System in some way.
Some of the momentum of incoming matter would be transferred to the Solar Systme, acting as a brake to the motion of the Solar System. The motion of the solar system relative to the local interstellar medium would slow, lowering the rate at which matter entered and the rate of deceleration. As the solar system slowed with respect to the local interstellar medium, its velocity with respect to the galactic center of mass would also change, changing the orbit of the solar system around the galaxy.
As mass accumulated in the rear end of the barrier and formed a new star system, the gravitational attraction between the new star system and the Solar System would pull them closer together, altering the motion of the Solar System relative to the interstellar medium and also to the galactic center, changing its orbit to a degree.
It is possible that as matter accumulates to form the new star system, its gravity will become stronger on the Solar System than that of any much more distant object. The two systems might be drawn closer to each other and collide with disastrous results.
Interstellar space might be said to begin at the heliopause, but the Solar System extends many times as far as the heliopause.
The solar system is believed to have an Oort Coud of billions of comets orbiting the Sun at distances both inside and outside the 1/6 LY or 10,000 AU radius of he barrier.
The Oort cloud is thought to occupy a vast space from somewhere between 2,000 and 5,000 au (0.03 and 0.08 ly) to as far as 50,000 au (0.79 ly) from the Sun. Some estimates place the outer boundary at between 100,000 and 200,000 au (1.58 and 3.16 ly). The region can be subdivided into a spherical outer Oort cloud of 20,000–50,000 au (0.32–0.79 ly), and a torus-shaped inner Oort cloud of 2,000–20,000 au (0.0–0.3 ly). The outer cloud is only weakly bound to the Sun and supplies the long-period (and possibly Halley-type) comets to inside the orbit of Neptune. The inner Oort cloud is also known as the Hills cloud, named after Jack G. Hills, who proposed its existence in 1981. Models predict that the inner cloud should have tens or hundreds of times as many cometary nuclei as the outer halo; it is seen as a possible source of new comets to resupply the tenuous outer cloud as the latter's numbers are gradually depleted. The Hills cloud explains the continued existence of the Oort cloud after billions of years.
So the barrier would let in comets from outside whose orbits were perturbed into highy elliptical ones by nearby stars. If those orbits had aphelions outside of the barrier, the comets would strike the barrier on their first outward leg of the new orbit. Depending on what happens when a comet strikes the barrier, astronomers on Earth might detect bursts of energy as that happens. And possibly different observatories detecting the same burst of energy could get parallaxes and determine that those hypothetical explosions are all 1/6 LY or 10,000 AU from the Sun.
And that is about all the effects I can think of.