I'm aware that I am restricted to working within my field (engineering)
Not exactly. "Engineer" status is more general than just engineering and covers all jobs requiring scientific or technical skills other than those specifically covered by some other status such as "Researcher". The legal definition is
Activities to engage in services which require specialized skills or knowledge pertinent to the field of physical science, engineering or other natural science fields [...] (except for the activities listed in the right-hand column of the "Professor", "Artist" and "Journalist" sections in Table (1), and the activities listed in the right-hand column of the "Business Manager" to "Instructor" sections, "Intra-company Transferee" and "Entertainer" sections in this Table) based on a contract entered into with a public or private organization in Japan.
The jobs you can do are precisely those that, in immigration's judgement, fit this definition.
would it be legal to take remote work from a foreign company?
The law says nothing about remote work; the general consensus is that immigration does not care about it, and so for immigration purposes, remote work from outside Japan is the same as no work at all. So on the one had, yes, it is legal (no matter the type of work) since it does not constitute working outside your status. But on the other hand, if you do only remote work, as far as immigration is concerned you are unemployed, and so your status may be revoked and even if it is not revoked you will certainly not be granted an extension.
Or what about start my own business?
Management of a business belongs to "Business Manager" status, and is explicitly excluded from "Engineer" as defined above, so no, you can't start your own business.