This is inevitably going to be somewhat subjective, but consider the following:

1. There was a time period in 1969-1970 where, for nearly a year, we had an attempting bombing or hijacking carried out by far-left groups every single day. During the Cambodian protests, professors who didn't fly peace flags from their houses often had their cars burnt, their houses attacked, etc, and Washington DC was essentially under siege. There were even cases of professors having their lifelong research destroyed in arson or bombing attacks. There was also a pretty nasty risk of right-wing backlash to this, as both Johnson and Nixon very well knew and took into consideration with their Vietnam policies. Wallace was just one prominent manifestation of this. There were times-most prominently in 1968 and in May of 1970-where it looked like the nation was having a nervous breakdown of sorts.

Thankfully, for a host of reasons, it didn't get as bad as it could have, not least because the much maligned Hoover-era FBI was pretty effective at shutting down really nasty far-left and far-right groups without killing anybody. (EDIT: At least for the most part, when you consider Fred Hampton.)

2. Whole swathes of cities were burnt down in the 1968 riots-two whole miles around Madison Street in Chicago, for instance. Some, like the area around 14th Street in Washington D C, didn't recover for multiple decades. Urban warfare was a very real possibility, and on a very limited scale, it did happen a fair amount in the late 1960s and early 1970s.

3. Aside from Vietnam, we were dealing with a lot of very nasty foreign policy problems, including a USSR that had just reached nuclear parity with the US and a deeply... "world revolution phase", isolated Cultural Revolution-era China (think basically the North Korea of the day). They nearly engaged in nuclear conflict in 1969. Plenty of trouble in the rest of the world-Middle East, stability of NATO, et all-as well. Back in DC, the whole consensus behind US foreign policy in WWII was collapsing in the late 1960s.

4. I'll let someone else speak for LBJ, but Nixon's paranoia about enemies stretched back to the beginning of his Presidency. Nor was it fully ungrounded in reality: even paranoids have enemies, and Nixon certainly had a lot of them. Nixon's blood feud with some of the "best" circles in the US stretched back to the early 1950s, and had only escalated throughout the years. This doesn't change the fact that Nixon took care to inflame those numbers by his own actions as a President, but as I've mentioned in a previous post here, the root of the conflict between Nixon and "the Establishment"-or at least certain sections of it-stretched beyond simple policy differences. Much like Bill Clinton and the swathes of the nation that hated him, there probably was nothing Nixon could have done to make them not go after him. Nixon and Clinton both certainly could have avoided inflaming the numbers of their enemies by not treating professional civil servants and related figures who might never have liked them or voted for them, but would have respected the office, as pre-emptive enemies: but the existence of their hard core gut cultural haters, the types that would have rather seen the nation suffer than them succeed, that was a fait accompli before they took office, and a degree of paranoia was probably just mere realism.

All in all, for all the above stated reasons and more, I'd say that the paranoia of people who had the ill luck of running the USA in the late 1960s was quite grounded in reality. I'm of the opinion that Nixon, Humphrey, RFK, Reagan, and Rockefeller all deserve credit for even wanting to be POTUS in 1968.

Sources:

Stephen Ambrose: Nixon, Volume II.

Rick Perlstein: Nixonland.

Even Thomas: Being Nixon.

Jussi Hanhimaki: Flawed Architect.

Dan Berger: Outlaws of America.

Dave Gergen's Memoirs.

EDIT:

Here's a story that will slam my point on home: the first thing Richard Nixon was given upon assuming office was a stack of forms from the Johnson Administration. All Nixon had to do was pen in the name of the city and his signature, and martial law would be immediately declared in that city. It really was that bad. '

LBJ mentioned in his memoirs that he considered it a miracle that there was not mass anarchy in the inner cities upon Nixon's inauguration and term in office.