Do not give up on liquid-fuel rockets! The first liquid fuel rocket engine was built in the Victorian era by Pedro Paulet in about 1895.https://en.wikipedia.org/wiki/Pedro_Paulet Honesty compels me to point out that NO corroborating evidence has been found for his claims, first made in 1927 (Believe me I have searched! I'll send you the documentation if you want). However, his engine should have worked, and would have an ISP of maybe 120-160. Of course, my hybrid (below) is much better.
You do not need turbopumps, a liquid fuel rocket engine can be pressure-fed. For a great example, see the Sea Dragon rocket. It was the largest rocket ever seriously proposed; please note that it was to be built out of steel, by a shipyard. https://en.wikipedia.org/wiki/Sea_Dragon_(rocket)
Also, the first ion-drive rocket engine was built by the Norwegian physicist Kristian Birkeland in 1912, and tested in an vacuum chamber.
I have been developing a thorough story, and have hundreds of inventors, characters, and methods. In my story, people go to Mars in 1917. I find that it would be very difficult to move this date back much more than ten years (twenty, tops), though, because so many enabling technologies were invented around this time.
I use a hybrid rocket engine using nitrous oxide and paraffin.