A common refrain in these kind of discussions is "there is no such thing as safe, only safer." Nothing we make is perfectly safe in an absolute manner - we don't design buildings to withstand asteroid impacts and we don't design cars to survive falling off a bridge. Instead, when we say safe, we usually mean "safe enough" - the level of risk is acceptable based on an understanding of all the ways we anticipate it could fail.
The hard part of this is defining the acceptable level of risk for a given situation. In many fields, a governing body has already defined the acceptable level of risk and written a code/standard/regulation that we get to follow. This saves everyday engineers and designers from having to spend a lot of time deciding what is and isn't acceptable. For people working in a field that doesn't have such standards, or people developing the standards, a risk assessment (however formal) is required to decide what risks can be tolerated. These assessments involve many factors, including:
- What are the consequences of failure? If your water is just warm enough to cause discomfort, you may be able to tolerate that risk. If it is so hot that it could cause permanent injury, your risk tolerance should be much lower.
- How likely is a failure? If the water is in an open container that frequently splashes on people, you should keep it at a cooler temperature than if it's in an insulated tank and could only escape in a catastrophic equipment failure.
- What are the options for mitigating this risk? If using room-temperature water just means that your experiment will take a little more time, it may be prudent to just not heat the water at all. If your experiment can only be done at high temperatures, then you might try to mitigate the risk by not allowing employees near the hot water, or requiring them to wear special insulating clothing.
So getting to a specific number depends on a variety of other aspects of your design. Certainly one conservative approach would be to follow whatever regulations are in place for tap water in your jurisdiction. Keeping the water in the neighborhood of 38 degrees celsius would be one reasonable option - about what is comfortable for people to touch. Of course if your process requires employees to keep their hands in the water for prolonged periods of time, even this may be too hot.
However if the water is piped and insulated and employees will not be exposed to it directly, you could consider following standards established for steam heat or perhaps industrial process piping if they more closely match your application. In broad strokes, these alternate guidelines will allow you to keep your water at higher temperatures in exchange for building more robust piping systems with additional safety features to reduce the frequency of ruptures.
Without more specifics of your application, system, and jurisdiction, we can't conclusively offer a specific number, but hopefully this gives you some pointers for finding the right value.