Can the landlord keep coming into our space?

In situations other than true emergencies, landlords may enter rental units only for valid reasons and with adequate notice. Most states regulate for the reasons for entry, when and at what time, and with what amount of notice.

The landlord's entry rights try to balance the competing rights of renter and owner. Valid reasons for a landlord to enter include maintenance and repair (indeed, if the landlord didn't perform maintenance and make repairs, he would be in violation of his obligations). Another common valid reason is to show an apartment or property to prospective renters, buyers, or lenders. Landlords can always enter in a true emergency—for example, if pipes rupture somewhere in the building, they can enter your space to shut the water off.

However, unless it's an emergency—a fire, a major leak, an electrical fault, and so on—in most states, the landlord needs to provide notice before entry. Some states specify the precise amount of notice—24 hours, 48 hours, and two days' notice is common. Other states simply require "reasonable notice" under the circumstances. In Ohio, 24 hours' notice is presumed to be reasonable; and landlords should enter only at "reasonable" times (such as not in the middle of the night). (Ohio. Rev. Code Ann. § 5321.04(A)(8).) Interestingly, in Ohio, landlords may enter to deliver packages that are too large for the tenant's mail facilities. (Ohio Rev. Code Ann. § 5321.05(9)(B).)

Your landlord's constant entries for maintenance seem to be excessive. Unless there's a true emergency with respect to any of the systems he's checking up on, these frequent visits are intrusive and violate your privacy. Have a talk with the landlord, and if he won't reform, consider invoking your remedies under Ohio law. By law, if the landlord's visits amount to harassment, you can recover actual damages resulting from the entry or demands, go to court to get an order telling the landlord to stop further such conduct, and obtain a judgment for reasonable attorney's fees. You may also terminate the rental agreement. (Ohio Rev. Code Ann. §5321.04(9)(B).)