Judging from the Tanakh, did ancient Jews also believe other gods like Baal actually existed? Or did the just believe they were fake gods, that people believe in but don't actually exist?
Looking at 1 Kings 18, where Elijah confronts the priests of Baal, priests that pray for a long time and get no response at all. Makes me believe the Tanakh portraits Baal as non-existent.
The Bible is clear that all other gods are false idols that only the misguided believed in, however many Jews fell into that category. So the answer is both. The Jews who worshipped idols believed in them, and the prophets of the Tanakh rebuked them for this erroneous belief.
Your question is: "Judging from the Tanakh, did ancient Jews also believe other gods like Baal actually existed?"
Your question if written in scholarly terms would be "Did ancient Jews believe in monotheism, or monolatry?" Monolatry is defined as: the worship of one god without denial of the existence of other gods.
Because of how your question is written the answer is we don't know and as of now no one can know for sure. We don't know because the Tanakh is not usually concerned with describing beliefs. You will find scriptures that portray other gods, or witches, or sorcerers as being real or having powers. And you will find scriptures that disregard anything outside of nature other than God. We also don't know because the Bible isn't a book, it's a library, which means it has multiple authors that span well over a thousand years of history.
If Book A of the Bible seems to describe other Gods as existing, but Book B doesn't; does that mean that Book B should be used to discredit Book A? Or is it that in Book A's time people believed differently than Book B's time? When Genesis describes as God walking through the Garden, but Deuteronomy says "behold you saw no shape or form of me on the mountain," which of these statements definitively proves what ancient Israelites thought about God's form? The answer is neither of them.
The only way to truly answer that question is to find other outside evidence from the time to compare against these books. First hand written evidence about beliefs would be great, but we haven't found any such documents. Archaeology would be second best, especially when compared to what the Bible says about Israelite practice during that time.
Based on the archaeology from within the land of Israel during pre-exilic times, especially when compared against Biblical narratives, monolatry seems likely to be the belief of Pre-Exile Israelites. This is not only because of the vast amount of idols we find all over Israel that date to this time period, but the types of idols we find. For example Asherah seems to be one of the most prominent idols, but she isn't usually worshipped alone. She is often described as being the wife or consort of Y-H-W-H, which is monolatry.
Further evidence for Asherah-worship includes, for example, an 8th-century BC combination of iconography and inscriptions discovered at Kuntillet Ajrud in the northern Sinai desert where a storage jar shows three anthropomorphic figures and several inscriptions. The inscriptions found refer not only to Yahweh but to ʾEl and Baʿal, and two include the phrases "Yahweh of Samaria and his Asherah" and "Yahweh of Teman and his Asherah." The references to Samaria (capital of the kingdom of Israel) and Teman (in Edom) suggest that Yahweh had a temple in Samaria...
Sources: https://en.wikipedia.org/wiki/Asherah#In_Israel_and_Judah https://www.penn.museum/documents/publications/expedition/PDFs/20-4/Meshel.pdf
Post exilic Judean belief seems to trend heavily toward monotheism. We start to get writings that have survived to the present day that often describe monotheistic practices or beliefs. Archaeological evidence shows idols as a whole tapering off. So a strong argument could be made that post exilic Judeans believed in monotheism and not monolatry.
But what we do know for certain is that after about the year 1,000 CE monolatry seems to stop existing within Judaism. We see this most clearly because of the writings we have from Jews describing "proper" beliefs such as Maimonides 13 Principles of Faith. The clarity of such writings, their focus on monotheism, and their widespread acceptance amongst Jews all over the world are the best proof we have of the vast majority of Jews no longer believing in monolatry, the same way vast majority of Jews stopped believing in exorcism and demon possession.
