Japan has a long and complex history of ranks and titles. Below is a very simplified explanation provided with the purpose of explaining two things: how the English language title of "baron" relates to Japanese titles and what ranks Oshima Hiroshi held in Japan.

The Japanese Imperial Court has awarded ranks, collectively called ikai (位階), for centuries but the number of ranks and their particularly stylings have changed over time. Furthermore, these ranks have also been tied to various civil servant positions or military ranks (collectively, kanshoku (官職)) at different periods of time. You can find some equivalency charts by searching 官位相当制 (kan'i soto sei). These ranks are numbered, with various grades within each rank. They are often stylized in English as "<senior/junior> <ordinal number> rank", for example, "senior third rank".

These court ranks were complemented by a separate peerage system as announced by Department of State briefing #307 on July 8, 1884 (Meiji 17). This system, called shakui (爵位), introduced five peerage ranks. These ranks borrow terminology from much older Chinese titles but are often stylized in English as roughly equivalent European peerage titles. I am unaware of any strict equivalency for each rank and have seen some variation in translation but for reference a common translation is: "duke" for koshaku (公爵), "marquis" for koshaku (侯爵), "count" for hakushaku (伯爵), "viscount" for shishaku (子爵), and "baron" for danshaku (男爵).

Oshima Hiroshi did not, as far as I can find, hold any peerage title (shakui), including the title of danshaku which is often translated as "baron". However, his curriculum vitae as submitted to GHQ court records in 1946 do show he held court rank. He was initially invested as the Senior 8th Rank (正八位) on December 1, Meiji 39 (1906) and rose through various ranks until reaching the rank of Junior 3rd Rank (従三位) in Showa 19 (1944) (I cannot make out the month and day but possibly January or February 1). Most of his court promotions coincided with or closely followed his Imperial Army military promotions.

In summary, it is not clear why Oshima was styled as a "baron" in English but his rise through the military ranks was followed by a similar rise through the imperial court ranks. His English language title may have been a nod to this rank, perhaps by his German contemporaries.

As an aside, since you asked what someone with the rank of "baron" might be the baron of, in short, nothing. As can be seen by those receiving investiture during the initial establishment, many of the recipients were former daimyo, most of which were also holders of imperial court ranks and positions. However, the establishment of a peerage system was, from some perspective, the only consolation prize after the abolishment of the han system (廃藩置県) in 1871 which was the end of feudal domains and required returning land holdings to the central government (版籍奉還). Although many of the daimyo clans were still wealthy, kept private estates, and may have continued positions of civil service, they were no longer lords of particular domains via hereditary right.