Is there a website that shows everyday how long you can be outdoors at your location before it becomes dangerously cold?

No: It is mostly a function of how you dress and otherwise protect yourself. Common sense and how you feel here. Well protected, there is no limit.

Outdoor Activity: Good day, The level of dress and the pace and duration of the physical activity can certainly extend outdoor winter time. Wearing multiple layers: silk long johns, multiple sock layers, tee shirts, head covers, hats and gloves can keep you toasty warm during a run or fast outdoor walk. Bring along a back pack to carry garments removed as you overheat.

No such thing: It all be different based on the intensity of the activity and also different on previous exposure to low temperatures, for example an Eskimo may be able to work at very low temperatures all day long, with minimal side effects, while an individual living in a tropical climate may succumb in hours. A website may give an idea about conditions, otherwise is common sense.

Not really...: There are many online weather resources as well as information on "normal" cold tolerance. "Dangerously cold" depends on your preparation for cold exposure. Read-up, talk with others, and plan in advance for your adventure; or if you're at home, look at the forecast. Stay well hydrated, dress in layers, and have fun :)

Doubt it: as so much depends on what you are wearing! Not even sure what you are asking-- if you have no clothes vs heavy prepared could change this answer from minutes to possibly days!