Lets take a purely biological approach here. (putting the psychological on the back burner)
What is stress?
Stress is a very vague thing that can mean different things to different people. We can all agree being robbed at gun point is a very stressful situation, however; some people find talking to others stressful while others actually find it relaxing... Since we're talking biologically stress can be defined as any situation where your body releases chemicals that are what result in the physical feels of stress (epinephrine being the big daddy of them) any situation be it physically, mentally, or emotionally stressful will trigger various degrees of stress chemicals to be released.
Examples of where stress turns bad
Stress in itself isn't bad, nor are the chemicals released in stress, however; prolonged or excessive release of these chemicals (extreme or chronic stressors) can have VERY serious heath implications.
Epinephrine for example causes blood platelets to secrete ATP which causes constriction of the veins. short term this increases the rate of blood flow (arguably a good thing) but in excess it can directly cause stroke and heart attack. (When someone is "scared to death" this is typically the cause)
Other stress chemicals have been proven to cause cancer, rheumatoid arthritis, severe damage to the immune system, reduction in endorphin release (IE your "pain tolerance" is reduced), increased ACTH which leads to increased violent behavior. (This is why when you are stressed you just want to hit things)
Some of these afflictions such as arthritis and damage to your pituitary gland are permanent generally speaking which is why excessive stress is literally killing you.
Where stress is good
Now I will note stress biologically is a good things. We feel stress for reasons typically that have been engrained into us from thousands of years of evolution and will serve us well to stay alive and healthy under normal circumstances. Epinephrine's fight or flight response has probably saved more lives than one could comprehend. ACTH has probably saved people in life threating situations by giving them extra drive to fight back against a threat they'd otherwise cower in fear from.
Now where does stress fit in well for the modern man outside of potentially dangerous situations? Can it still be a good thing?
Absolutely! Stress in moderation is very healthy, and can lead to a feeling of accomplishment when the cause of the stress is overcome.
But I'm focusing on biology so...
Epinephrine, that thing that causes stroke and heart failure in excess. Well let's take a normal amount, say a friendly soccer game. You're generally relaxed, but your body will be under at least a little stress physically for such a taxing activity, this means you're getting a nice little trickle of Epinephrine causing just a tiny bit of constriction in your veins. This makes your heart work a little harder, but not so much it can't handle it. Your heart is a muscle, and like any muscle it needs exercise, otherwise when it takes a lot less to do something that can be too much for it to handle.
The chemicals that lead to rheumatoid arthritis also help the body break down and replace damaged tissue. You want to keep a little of this moving around to help keep that regenerative cycle working like clock work. It'll help your body heal up properly from injuries better.
The chemicals that cause damage to your nerves and immune system actually help kick those parts of your body into high gear. In other words a healthy amount of stress actually helps you focus as well as helps your body fight off infection. (don't stress yourself out when your sick, that's a too little too late situation) Note: this focus is probably not the lion's share of why you focus on manageable tasks, that's more likely to non stress related self preservation than stress itself.
Ultimately a little stress though is good if not for any other reason than that your body is better prepared when an extremely stressful situation rolls by.
How much stress is healthy?
This is VERY disputed and arguably varies person to person. Generally speaking though if the stress is enough you feel like punching someone in the face, your chest is sore, or you have an unusual numbness or ache in your hands, feet, or head you went way past the healthy point. (I will also say if you have any of these symptoms other than the urge to hit someone you should mention it to your doctor, it could be signs of a serious and not yet diagnosed health situation. Especially the chest.)