The Secret Service did not start protecting the president full-time until after the assassination of William McKinley in 1901. The president would travel in public before this time with only a few aides or associates as a limited buffer against the public; in 1876, President Rutherford B. Hayes traveled to the exhibition in Philadelphia as a civilian alone. Five years later, President James Garfield was shot by Charles Guiteau at the Baltimore and Potomac Railroad Station in Washington. Garfield was traveling by train to give a speech at Williams College and he was surrounded by the Secretary of State and Secretary of War and a few Washington city police officers.
Candice Millard (2011), Destiny of the Republic.
Is the increase in Presidential security in any way related to the increase in mass media, which I would imagine made the President more recognizable to the general populace?
Since your question has already been answered from a security standpoint, I will try to handle it from a cultural one.
I remember learning that when Washington became president, there was a good amount of debate and nervousness among American officials, as a democracy had not been implemented on such a scale since the Roman Republic, and individuals held varying opinions on how the president should act and what level of professionalism and separation from the people he should hold.
You must remember that since there was no precedent set, Washington had an enormous amount of pressure on him. He likely realized that future presidents would emulate his style of ruling and his approach to the balance of power, particularly when it came to the left vs right wing debate. This is also the reason why he often took advice from both Jefferson (a left-wing individual) and Hamilton (a right-wing one), and both of which were constantly trying to sway Washington to their side.
But I digress. Washington eventually decided that the best decision was to be less formal and regally dressed than a monarch, but moreso in both regard than the average citizen. So, no, there was no enormous cultural "barrier" that separated you from the President as it would from, say, the Holy Roman Emperor, and you would indeed be permitted to speak with Washington.
However, George Washington stopped being president in 1797, and was succeeded by Adams; your question asks specifically for 1798. I cannot speak for Adams, however I am willing to assume that his approach was not particularly unlike Washington's, especially given that the White House remained open to the public for decades after that.
In fact, Lincoln even has a quote about the sheer number of citizens that came to visit him in the White House:
"For myself, I feel—though the tax on my time is heavy—that no hours of my day are better enjoyed than those which thus bring me again within the direct contact and atmosphere of the average of our whole people." (Source: www.whitehousehistory.org/abraham-lincolns-white-house )
I hope this helps to answer your question.
Another source:
Can I ask a follow-up question about the public visiting their leader? Has it traditionally been possible in Western societies to do that?
It's not quite 1798, but fast forward to 1801 and Jefferson was approachable. He often answered the White House door in robe and slippers. His social style in the White House on occasion left visiting European ambassadors and dignitaries a bit aghast, and occasionally feeling downright insulted, as he would approach and introduce himself without pomp and decorum. This informal, "crass" breech of noble etiquette was a deliberate departure from customs of royalty and power common in Europe.
Sources: Dumas Malone, Jefferson and His Time (1848-81)
Joseph J. Ellis, American Sphinx: The Character of Thomas Jefferson (1996)
Fawn M. Brodie, Thomas Jefferson: An Intimate History (1974)
[The Thomas Jefferson Hour] (http://jeffersonhour.com/) -- an engaging weekly podcast with the 3rd President (as portrayed by Clay Jenkinson)