I read this when I was a young teenager in the late 60s so my memories are faint, but I would love to read it again. What I do remember: Humanity is spreading through the solar system thanks to the invention of inflatable DIY spaceships. The ships resemble bubbles and are called "bubs." When people want to say "Be cool," they say "Be cavalier." The spacesuits are Archer models -- Archer 7, Archer 8, etc. Quite a bit of political intrigue, some violent. The UN patrols the space lanes.
The Planet Strappers (1961) by Raymond Z. Gallun...?
THE PLANET STRAPPERS started out as The Bunch, a group of student-astronauts in the back room of a store in Jarviston, Minnesota. They wanted off Earth, and they begged, borrowed and built what they needed to make it.
THE PLANET STRAPPERS got what they wanted -- a start on the road to the stars -- but no one brought up on Earth could have imagined what was waiting for them Out There!
From a review:
The book opens in purest YA hard SF mode, with a group of college-age kids trying to make their way into space. It seems that most of the trick is to acquire a space bubble, or "bubb", which is what it sounds like -- not much bigger than person-sized, a bubble based on a super-plastic, with plenty of air production capacity, in which someone can survive for a pretty significant time in empty space. The group includes a diverse(-ish) mix -- one woman, a rich kid, a handicapped kid, a couple of football star twins, a delinquent, a couple more, plus the eventual viewpoint character, Frank Nelsen, the straight-arrow honest type. This first part goes on for 50 pages or so, pretty effectively, as the "bunch" (their name) navigates such issues as making the "bubbs", learning how to use them, passing the required space fitness tests, raising the needed money, and so on. The girl member drops out, realizing the the regular Space organization is desperate for women, and another guy washes out for lack of psychological fitness, and one member has to deal with his mother who won't let him go.
Finally they head into space, and things get a bit stranger from that point. Frank ends up in a terrible situation on the Moon with a murderous fraud. One of the dropouts ends up a stowaway, causing even more trouble. Eileen, the girl member of the Bunch, becomes fairly successful, as what seems to be a Madam, though the book is too YA-oriented to go into detail about that.
The book continues, becoming something of a travelogue through the Solar System. Frank's goal ends up to be establishment of a free space-based set of habitats, based on the "bubbs", and in the end to make a home that the girl back home he's sweet on can come to. But he must deal with a lot of problems on the way there, most importantly the issue of space pirates ... a group which seems to include the one washout from the original "Bunch" who had stowed away. One of the other members becomes a legendary explorer, eventually heading to the Outer Planets solo in a bubble. There are deaths among the bunch, and failures, but by the end we see a portrait of the establishment of a new frontier.
Searching the Google Books preview, the phrase "be cavalier, fellas" is used, and the word "archer" comes up plenty of times.