We are taught how important it is to study Torah constantly. That the entire world rests upon those who learn Torah. So how is it possible to ever take time off from learning in order to do other important things in life like being with family, staying healthy, seeing G‑d’s world, etc.?
This is a very important matter to address, and it’s great that you asked the question.
Truly, there is nothing more important and beneficial for the world than learning Torah. As you make reference to, the continuation of the world is dependent on Torah learning (Pesachim 68b). Before the Torah was given at Sinai, the world was in a state of limbo, contingent on the Jewish People’s accepting of the Torah (Shabbat 88a). Torah learning is described as being one of the pillars upon which the world is supported (Avot 1:2).
We are commanded in the Torah, “You shall be engaged in the words of the Torah day and night” (Josh. 1:8). Our sages rhetorically taught (Menachot 99b) that if there is a time which is neither day nor night, you may desist from learning Torah. Of course, since there is no such time, the intention is that one must learn constantly. [By the way, dawn and twilight combine both day and night, which results in an extra obligation to learn at those times! (Sh.A., O.Ch. 1:1, M.B. 2, from Shelah)]. In fact, since the existence of the world depends on Torah, it has been suggested that one of the reasons for the dispersal of Jews across the globe is to ensure round-the-clock Torah learning.
There have always been select individuals who were so connected to G‑d and so devoted to Torah that they literally learned Torah nearly all day and night every day and night for most of their lives. These great and pious individuals are referred to as one whose Torah is his sole pursuit in life, “Torato omanuto”, whose special status even exempts one from other very important mitzvot (Berachot 8a).
However, while all of the above represents the ideal, and such rabbis are worthy of emulation, this is not expected to be the reality for the vast majority of people.
For example, a person needs to eat and sleep. Certainly at these times he cannot learn Torah. The reason he is allowed to take such time-outs is in order to restore and replenish his health and strength so that he’ll be able to continue studying afterward. This is also true of earning a livelihood. One is exempt from learning in order to be able to support himself and his dependents, so that all may engage in Torah in the long run. And the same is also true regarding taking care of other needs which maintain and ensure his physical, emotional and mental health and balance.
This dynamic of taking time off from learning in order to take care of one’s needs was succinctly referred to by the Sages (Menachot 99b) as “its nullification is its fulfillment” (“bitula zehu kiyuma [yesoda]”) insofar as the Torah permits, and recognizes the value of, taking away from learning in order to be able to continue learning.
In fact, since all productive, permitted activities are governed by Torah teachings, and particularly if one has the intention to rest in order to learn, these various time-outs can actually be considered as continuing one’s Torah learning, albeit in different forms. In this vein, the Kotzker Rebbe offered a most brilliant explanation of the teaching that one who interrupts his learning in order to note the beauty of a tree is as if he is liable for his life (Avot 3:9). He remarked that anyone who considers appreciating the wonder of G‑d’s Creation as an interruption of Torah rather than a fulfillment of it, is forfeiting his life!
The same applies for taking time to appreciate and partake of other forms of G‑d’s blessing like the other ones you mention (family, health) and more, as long as it is within reason and with the proper intention.