The issue is not really whether the road or path has a barrier, it's about if access serves a public purpose. Private roads do not become publically accessible by reason of the fact that a postman or milkman has an implied invitation, on that road or land, so on the whole, you are a visitor on a private road. So you could establish that an unadopted road, without any physical barrier to or for access, is still not a public place.
A public place can be defined as a place "the public have access or which members of the public are to be found without having obtained access either by overcoming a physical obstruction or in defiance of prohibition express or implied." However, the extent and nature of the public's reason and frequency for accessing the road is what it boils down to, not any barrier.
Probably best summed up:
Planton v DPP
The case of Planton was a prosecution for driving in a public place vehicle with excess alcohol. The case turned upon the definition of a 'public place'. Access to the causeway from the mainland was not prevented by any physical obstruction.
He was charged under section 5 (1) (a) of the Road Traffic Act 1988. Section 5 (1) (a) provided;
“if a person drives or attempts to drive a motor vehicle on a road or other public place after consuming so much alcohol that the proportion of it in the breath, blood or urine exceeds the prescribed limit he is guilty of an offence”
The prosecution alleged that a road linking an estuary island to the mainland was a public place.
The causeway allowed access to and from the residences on the island.
Royal Mail collected and delivered to the island.
Signs present on the road were "Private Road, Residents and Authorised Vehicles only" and "Private Road, No Parking".
Lord Justice Pill gave consideration to the question of whether there had been any evidence of public usage. He found that the evidence in the case was
"wholly insufficient to entitle a court to conclude that the causeway was a public place"
"There is no evidence of any general public access to the island for any purpose. If, for example, there had been evidence of a viewpoint on the island from which there was a beautiful view of the sunrise, and evidence of use by the public of the causeway to obtain access to the island to enjoy the view, that would be evidence material for present purposes"
http://www.mibclaim.co.uk/resources/library/cases/planton-director-public-prosecutions-2002/
In a nutshell, unless the general public have a reason or entitlement for access to a private road, it's likely not be construed as a public place.
