We've specified tankless water heaters on some jobs for a decade or so. There are issues, but overall they are okay. Whether gas or electric the tankless must be sized much larger because it must heat water so much more quickly, so electrically you'll definitely need a new circuit. If you have an old house, or if the water heater is located remotely from the panel with the wiring running through finished and inaccessible space, or if it's an old house with limited electrical capacity, this may be a problem. This will also be an issue if you have a generator, as it takes a much larger generator to run a tankless water heater than a conventional storage model which heats water much more slowly. You may also need a larger line for gas models, although usually not.
Sizing is essential; you need to know not only your total flow requirements, but also your minimum ground water temperature. Part of sizing is also what you are willing to accept; do you have to have sufficient capacity to run the dishwasher or washer as well as the shower? How hot is hot enough for you? Are you willing to accept less hot water volume? You also need to know if your dishwasher and washer will handle cooler water before you make those decisions; newer ones with electronic timers can accept any temperature of water, but with older models with mechanical timers accepting cooler warm water may not be an option as they are designed assuming a minimum hot water temperature and have only a mechanically controlled time to heat it to the required temperature.
You also need to know if you are on well water or treated municipal water. The latter almost always has chlorine or chloramine added, to kill bacteria and amoebae*. But with well water (or otherwise untreated water) you may well be taking in bacteria or amoebae that can potentially kill you. In that case, you absolutely need to heat your water to a very high temperature, as breathing this things is, as Mark R said, potentially fatal. In that case, you have to have mixing valves to make your house child-safe. 60 C hot water and mixing valves are always the best design - they aren't common on houses in many areas because of the cost - but it's essential for well water. But even the best municipal water can sometimes be low on chlorine/chloramine, especially reservoir supplies during hot, dry summers.
You may or may not notice much savings over a newer storage tank water heater. Newer storage tank water heaters are much better insulated than are older models. If you use a lot of hot water, your tankless water heater won't pay off very quickly. If you use a lot of hot water and are currently limited by the hot water running out, your tankless water heater may not ever pay off, especially since you'll tend to use more hot water than before. If you use little hot water, your tankless water heater may pay off fairly quickly because you aren't heating water and then keeping it heated for hours before using it. A new storage tank water heater on a timer might then be an even more practical choice; the trade-off is its lower initial cost versus the tankless water heater's ability to respond instantly to a change in schedule. (If you normally use no hot water before seven AM and then one day get up at three, a timer-controlled storage tank water heater may have only luke-warm water; a tankless water heater will have its normal unlimited hot water.)
*Really, all protozoans.