Is it currently possible to build a jet engine with 1,050 kN of thrust? (236,049 lbf)
The limit to the size of turbomachinery is considerably larger than the size of current aero engines. 

To estimate how large cores can get, we can look at power plant SCGT. You can get heavy frame turbines as large as 576 MWe for the SGT5 (this is half of a typical nuclear reactor at 1000 MWe). The aeroderivative LM6000 delivers 40 MWe out of a 270 kN turbofan core, and the largest aeroderivative, Industrial Trent 60, provides 60-70 MWe out of a 415 kN thrust turbofan core. 

This would suggest an thrust equivalent of 3.5-4 MN or 800,000-900,000 lbf for the SGT5. You can't just fly a frame turbine like this, except in the hold: it's slow to spin up, its speed is on the low side, and it's not certified to aviation standards. But that's just a matter of design. It simply shows the amount of power that can be crammed into a single turbine core. 

So, could we make a 1 MN engine? Yes, we are making cores with several times the power. Could we make an airworthy 1 MN engine? With effort, perhaps, of course it would cost money to develop. Would it be as good as the engines we have? That's the million dollar question. Or it would be, and more of a billion dollar one, if there were a market for it.