OK, I'm gonna take a crack at this. Everything I postulate is based on the year being 1903, and the country in question being the USA. Handwaving away everything beyond the original question, I'll just focus on "technical sophistication". The task, then, is to build a "Land Monitor", since monitor is really more appropriate than "battleship", for what a tank is.
OK, so we've got tracks. What's next?
Part 2: Engine
So far in this thread, there has been a lot of discussion about engines available in 1903. Renault, Daimler, Mercedes, etc. Gasoline versus steam, how many horsepower you need to move a machine. Problem is, they've all missed an important candidate for 1903 "Land Monitor" power. I present to you the most powerful self-propelled internal combustion vehicle in production in the USA in 1903:
This is the Hart Parr No.3, later known as the 17-30. As early as 1901, Mr. Hart and Mr. Parr had built their prototype No.1 tractor, based on their successful line of gasoline stationary engines. By 1903, they were producing the No.3 machine, which was the first serial-produced tractor in the USA. The big cylinder sticking up in front? That's the radiator - the smokestack looking part is actually where the engine exhaust was piped, exiting up to create a draught to pull air through the cooling system. The three-spoked wheel in the upper middle? Clutch assembly. The machine featured a fully enclosed crankcase and fully pressurized lubrication. If you look closely, you can see the gears of the transmission and final drives, as well as the worm gear for the steering system. It weighed 14,000 pounds, and was equipped with a two-cylinder, 1654 cubic inch (21.7 Liter, for the metric set) gasoline engine that ran at 300rpm. The engine produced 30 shaft horsepower, and the entire machine had 17 drawbar horsepower.
Doesn't sound too impressive, does it? Well, the numbers don't lie, but they also don't tell the whole story. First, this is an agricultural application, and that 30 shaft horsepower refers to continuous power. Hook this machine to something that needs to spin - a threshing machine, pump, sawmill, rock crusher - and it will make those 30 horsepower all day, every day, for as long as you feed it fuel, oil, and water for the radiator. Even as early as the 1900's, manufacturers understood the advertising appeal of horsepower, so a 35hp Renault probably could make that number for a brief amount of time, and could only sustain a portion of that in continuous operation.
The 17 drawbar horsepower number? That one is even more deceptive. Drawbar horsepower is just that - how much pulling power does the machine have, expressed in horse-equivalent units. Nowadays, that number is one calculated based on engine power, gearing, traction efficiency, etc. But in 1903, it was a direct equivalent. So, this machine had enough ooomph to pull the same load as 17 horses hitched together - in addition to it's own 7 ton carcass. Again, not in spurts or with a running start. From a dead stop, and then for as long as the fuel, oil, and water holds out. Or until the machine breaks down.
Consider this. There is a 1903-built No.3 machine in the Smithsonian that was bought back by the Hart-Parr/Oliver company for their own collection. In 1926. It was still in daily use when it was bought back, 23 years later. Most of these early pre-WW1 tractors were in use well until the 1920's and 1930's, and it wasn't until the massive scrap drives during WW2 that they became rare. These machines were sophisticated in their simplicity, and designed to work and keep on working.
The No.3 was designed for drawbar work, meaning it was geared to pull things. It had a multi-speed transmission (3 speeds, I think?) and had a road speed of maybe 4 miles per hour. So, not fast - but running on steel wheels with no suspension of any sort at all, 4 mph is about as fast as you want any machine to go. You wouldn't think it, but trust me - things can happen awful fast at a brisk walking pace.
Now, look at the picture again. Squint a bit, and with your imagination remove the wheels and substitute in a set lengthened Lombard-style track units. Maybe swap the radiator position with the drivers position, so he can see where he's going. Doing this will cut out a lot of the plumbing going back and forth, and if you lose the canopy above you have a lower silhouette overall. Can you see it? We've created the fully tracked crawler tractor, 10 years earlier at least. It's going to be heavy, probably around 16,000 pounds or so when the tracks and steering clutches are installed. It will be slow, since we'll really not want to gear it faster than 5mph or so given the lack of an sort of suspension. But it will be relatively reliable, moreso than most passenger cars of the period. It will be powerful enough to haul itself around, plus whatever extra you hang on it - after all, it has enough tractive power to move 17 horses worth of load. All of this with extant 1903 technology. Not stuff that can be built with 1903 technology, but with actual machines being built in 1903.