My first question is what happened to the original flue? Was it completely removed? If so, the venting of the water heater should have been part of the plan to begin with.
There is never a scenario where you don't have to vent a gas water heater to the exterior.
It might be possible to use a vent inducer fan to vent out a side wall, IF the manufacturer specifically allows it for that model.
A vent connector can go horizontal, BUT the vent can only go 1.5 feet very every inch of diameter. Most Cat I water heaters have a 3" connection to the draft hood, so a 3" vent pipe can only travel for 4.5 ft. Upsizing the vent to 4" will allow it to travel 6 ft. (and always with a min. 1/4"/ft. of slope back to heater). The vent connector cannot travel more than 75% of the height of the flue. There are restrictions on the vertical vent, similar to what Frodo drew. That is the code part, noting that the details are not that simple.
In addition, when a flue passes through a wall, floor or roof, it must be kept a minimum distance from combustibles, usually by using a metal shroud. It must extend at least 3 ft. above the roof, and at least 2 ft. above any obstruction within 10 ft. horizontally. For a 12/12 pitch, that would require that the flue extend 12 ft. above the roof (if more than 12 ft. from the peak). If the pipe were to extend up past the eave of the roof or were to exit the roof near the bottom of its slope, the flue pipe will be subject to damage from snow/ice. The flue piping must be at least B-vent piping, which is an aluminum pipe within a galvanized pipe. B-bent pipe is NOT insulated. When it passes outside in a cold climate, the water vapor (which is more than 9% of the flue gases) will quickly condense, which will corrode the flue pipe and cause other problems. (OP does not indicate what part of the country he is in).
Bottom line is that re-venting the appliance is not simple to meet the code requirements. I agree that you should replace it with a direct-vent or power vented water heater OR a heat pump water heater.