Why is it right to conclude that time and space are really infinite and continuous rather than discrete and discontinuous? — Nils Loc

Because there is no start or end in other of them. Neither any point in the middle. At least we cannot define any of them, therefore we cannot assume that they exist.
Heraclitus has given the best definition of time --even if the didn't use the concept of "time" itself-- 2,500 years ago, with his famous "Everything flows". This shows the continuous aspect of time. Then, he also descibed change --a basic element in the concept of "time"-- by saying "No man ever steps in the same river twice". This also shows the continuity of things. (Of course, a river has a start and and end, but this is besides the point of the analogy.)

If you try to think of a finite, descete piece of time or space, you would most probably think something that has to do with a period of a year, month, day, hour, etc. (time) or a distance in kilometers, meters, centimeters, etc. (space). However, all these measurents are arbitrarily set and are only indicative. They do not really exist as such. Neither a year nor a kilometer exists as such, i.e. physically. You cannot perceive them with any sense. They are concepts. They are conventions.

Defining and measuring time and space are used for description, explanation and comparison purposes only, as in geometry. We can define a point in geometry. But it will exist only "on paper". A point does not really exist. (What is its dimension? Can it be measured?) So, if it doesn't really exist, then "the distance between points A and B" does not exist either. That is, a distance does not really exist. It is a concept used for description, explanation and comparison purposes. And the measurement units and their amount that we use to defiine a distance are also arbitrary and may differ from one place or system to another. Moreover, we can never guarantee their precision: there's no perfect measurement tool either for time or space. In fact, the measurement of a period of time or a distance can never be so precise that we can consider it as "absolute". Measurement tools are created by humans and nothing created by humans can be perfect.

Either of these abstract properties are just mathematical inventions/conventions which prove to be useful. — Nils Loc

Right.

Time and space can obviously be divided (measured in units) — Nils Loc

I talked about "measurement" above.

If time is infinite, it's still divisible by seconds in relation the diurnal or lunar cycle. — Nils Loc

It's always about "measurement".

If space is infinite, it's still divisible by length of feet in relation to how much horse food, water or minutes it takes to get to town. — Nils Loc

It's always about "measurement".

So what am I missing? — Nils Loc

I can't really say. But from what I can easily see, you are using "measurement" as an indication and/or proof of the finiteness and discreteness of time and space. This is a very common mistake or, better, an illusion. We meet all kinds of "measurements" of time and space in millions of things everyday in hour life since eons ago. So they have become substitutes of the concepts of time and space themselves ...
I believe that ancient people, the lives of whom were much simpler and without such a multitude and amount of measurements, had a better notion of time and space!

Neither a year nor a kilometer exists as such, i.e. physically. You cannot perceive them with any sense. They are concepts. They are conventions. — Alkis Piskas

It's odd to assume time and space are really infinite and discontinuous on one hand, then deny the existence of a kilometer. If we're to be consistent here, time and space don't exist either. Personally I find units of measurements far more tangible and intuitive than any applied notion of infinity.

My quibble is pointless though. Thank you for your responses.

I believe that ancient people, the lives of whom were much simpler and without such a multitude and amount of measurements, had a better notion of time and space! — Alkis Piskas

Here I would assume any ancient people's notion of time and space was conditioned first by everyday discrete objects and relevant measurements that became standard to help them in their everyday lives. Zeno's preoccupation with infinity wouldn't be ordinary in any sense, and therefore wouldn't be suggestive of a common people's notion of time and space. But I don't really know.

It's odd to assume time and space are really infinite and discontinuous on one hand, then deny the existence of a kilometer. — Nils Loc

When have I denied the existence of a kilometer or other unint of measurement?
I feel you didn't get anything from all that I said.
And I have already said much.

Neither a year nor a kilometer exists as such, i.e. physically. You cannot perceive them with any sense. They are concepts. They are conventions. — Alkis Piskas

Yes and to be consistent you apply this also to the notions of space and time, infinite and continuous. They are concepts. They are conventions. With this in mind, what supports the belief that space and time are really infinite and continuous rather than finite and discrete?

Because there is no start or end in either of them. Neither any point in the middle. At least we cannot define any of them, therefore we cannot assume that they exist. — Alkis Piskas

There is no start or end to time and space. I'm just saying that this attribution is as imaginary/arbitrary as any contrary claim, that there is a start or end to time and space. Not a bid deal, just a silly quibble. It just occurred to me as a possible inconsistency on your part, but maybe I don't understand you.

Read it again, more carefully this time, paying attention to the emphases that I have added: "Neither a year nor a kilometer exists as such, i.e. physically. You cannot perceive them with any sense. They are concepts. They are conventions."
That's all I can do. And I'm sure that you can get the point.