TL;DR: There are no instrument procedures into 26MA, so there are no instrument procedures out of it either. Proceed with extreme caution. Departure in non-VFR conditions is not recommended.
As @757toga mentioned in a comment, the FAA only conducts a "diverse departure assessment" at airports for which it has responsibility to design instrument procedures.
FAA JO 8260.46, 2–1–1a
(1) Where IFR departure procedures are authorized by the approving authority conduct a diverse departure assessment for those airports with approved instrument approach procedures. ODPs are developed by FAA Service Providers at locations where they have instrument procedure development responsibility.
(2) ODPs may also be required at private airports where the FAA does not have instrument procedure development responsibility. It is the responsibility of non-FAA Service Providers to ensure a terminal instrument procedures (TERPS) diverse departure obstacle assessment is accomplished and an ODP developed, where applicable.
ODPs and/or non-standard takeoff minimums are developed "when obstructions penetrate the 40:1 departure obstacle clearance surface (OCS) as described in Order 8260.3." From a brief look at Google Earth, you are definitely correct that this is an airport which would qualify; even setting aside the trees, there is a hill approximately 4750 feet from the departure end of Runway 24, for example, and that hill is 300 feet higher than the field elevation at 26MA. The FAA's standard takeoff minimums presume an obstacle rise of no more than 152 feet per nautical mile from the runway; this hill is 2.5 times higher than that!
However, if neither an FAA Service Provider nor a non-FAA Service Provider has developed an instrument approach procedure for 26MA, then no one has conducted that diverse departure assessment and created non-standard takeoff minimums. This means there is nothing you can rely on but your own knowledge of the area when deciding whether to depart from such an airport under Instrument Flight Rules and in Instrument Meteorological Conditions.
It is not ATC's responsibility to know this information nor to police your activities, which is a common theme throughout JO 7110.65. ATC clearances are predicated on known and observed traffic; the pilot-in-command is presumed to have all appropriate certifications and currencies, to be flying a properly certificated and airworthy aircraft, to be operating in compliance with the applicable FARs, etc, etc. Even if an ODP does exist at an airport, ATC will not assign it unless it is necessary to ensure separation (JO 7110.65 4–3–2c2), and unless it is assigned "compliance with such a procedure is the pilot's prerogative."
So the clearance you would receive from Boston Approach would be the same as the clearance you would receive when departing any other Class G airport, whether public or private, and would—in accordance with 4–3–2c1(c)—sound something like this:
November 12345, cleared to [destination] airport as filed. Maintain three thousand, expect five thousand one-zero minutes after departure. Departure frequency 124.9, squawk 1111. [If necessary: When entering controlled airspace, fly heading 260.] Hold for release.
But this is assuming you would be issued a clearance off-the-ground at all. Note how close 26MA is to ASH, and to the ASH Class D surface area, and note how (according to the Skyvector link you posted) the airport has been analyzed to be "NOT OBJECTIONABLE IN ACCORDANCE WITH VFR." I don't want to say it would be impossible for you to get an IFR clearance while still on the ground at 26MA; nothing is impossible. But I could easily imagine it being unlikely.
For completeness: If you did take off from 26MA and attempted to pick up your IFR clearance airborne, you would need to either be at-or-above the minimum vectoring altitude when receiving the clearance, or you would need to confirm—per 4–2–8d—that you are able to "maintain your own terrain and obstruction clearance" until reaching the MVA, even if you may go IMC before that altitude. Obviously this is something that you should only agree to if you are very confident of your location and the location of any nearby terrain and obstructions, and the performance capabilities of your aircraft. It would not be the recommended Plan A.
The difference between getting your IFR clearance while you are on the ground at an airport and getting it after you have departed is, of course, the assumption that the area around the airport has been analyzed and an ODP or non-standard takeoff minimums have been published. That assumption does not hold true in this case, but the 7110.65 does not take account for that.
CONCLUSION: Although I do not believe this is a requirement per 14 CFR Part 91, the smartest thing would be to apply the "can I file IFR to this airport" rule in reverse. Namely, as you alluded to, the safe move would be to only attempt a departure from 26MA—or any airport which does not have a published instrument approach procedure—if the ceilings are no lower than the local Minimum Vectoring Altitude, meaning you can climb visually to that safe altitude. ATC procedures do not make distinctions between airports that are and are not served by instrument procedures and you cannot rely on ATC to provide a sanity check in this situation.