R, 1042 bytes

The credit for this answer should go to digEmAll, who did 499(!) bytes better than my version (left below). Most characters are printed by using the octal representation, e.g. cat('\041') prints character number 41 in octal, or 33 in decimal, i.e. !. There are a few additional tricks, explained in my original answer below.

cat('','')
cat('\041')
cat('\042')
cat('\043')
cat('\044')
cat('\045')
cat('\046')
cat("\047")
`?`=cat;?'\050'
`?`=cat;?'\051'
cat('\052')
cat('\053')
cat('\054')
cat('\055')
cat('\056')
cat('\057')
cat(+F)
cat(+T)
cat(1+1)
cat(2+1)
cat(3+1)
cat(4+1)
cat(5+1)
cat(6+1)
cat(7+1)
cat(8+1)
cat('\072')
cat('\073')
cat('\074')
cat('\075')
cat('\076')
cat('\077')
cat('\100')
cat('\101')
cat('\102')
cat('\103')
cat('\104')
cat('\105')
cat('\106')
cat('\107')
cat('\110')
cat('\111')
cat('\112')
cat('\113')
cat('\114')
cat('\115')
cat('\116')
cat('\117')
cat('\120')
cat('\121')
cat('\122')
cat('\123')
cat('\124')
cat('\125')
cat('\126')
cat('\127')
cat('\130')
cat('\131')
cat('\132')
cat('\133')
cat(intToUtf8(92))
cat('\135')
cat('\136')
cat('\137')
cat('\140')
`c\141t`('\141')
cat('\142')
`\143at`('\143')
cat('\144')
cat('\145')
cat('\146')
cat('\147')
cat('\150')
cat('\151')
cat('\152')
cat('\153')
cat('\154')
cat('\155')
cat('\156')
cat('\157')
cat('\160')
cat('\161')
cat('\162')
cat('\163')
`ca\164`('\164')
cat('\165')
cat('\166')
cat('\167')
cat('\170')
cat('\171')
cat('\172')
cat('\173')
cat('\174')
cat('\175')
cat('\176')

Try it online!

R, 1541 bytes

Most non-alphanumeric characters are of the form cat(intToUtf8(33)). The intToUtf8 part can be made shorter for most alphanumeric characters. The main issue is the characters cat(), which I will handle last.

For digits, simple operations like cat(1+1) work.

For letters, the shortest seems to be to use the constants letters and LETTERS (the alphabet) whenever possible, e.g. cat(LETTERS[1]) for A. This doesn't work for the characters LETRSletrs, for which we move to the slightly longer cat(toupper("l")) or cat(tolower("S")). For elr we have to revert to cat(intToUtf8(101)) and the like.

The code for the space can be made shorter with cat('',''), because cat will by default add spaces between strings (here two empty strings).

The brackets () are annoying: not using them seems to disallow calling functions. Here I used a standard R golfing technique, redefining some unary functions which have special syntax: cat(intToUtf8(40)) becomes

`?`=cat;`!`=intToUtf8;?!40

Finally, the letters cat. We need the cat() function, otherwise R will add some fluff; for example letters[1] or print(letters[1]) outputs [1] "a" instead of a. We'll need two distinct workarounds.

The function write is usually used to write to a file, but can be made to write to STDOUT: write(letters[1],'') works for a and c. Unfortunately, that still doesn't work for t. I almost gave up here...

Let's take a step back. The reason R adds [1] by default when printing is that most objects are of a vector type (here, a vector of length 1). The main exceptions are functions; in particular, calling for the body() or args() of a function doesn't add anything, but I couldn't find a way to use this. There are however internal non-vector types, such as promises, expressions and symbols (the latter are also called names), so a solution is to use as.name: as.name("t") converts the string to an object name, and outputs it at just t. I also used this for c, since as.name(letters[3]) is shorter than write(letters[3],'') (but we still need write for a).

We now need to find a way of creating the string "t", but cannot use letters, tolower or intToUtf8. The best I could come up with is as.name(rawToChar(as.raw(116))).

Full list:

cat('','')
cat(intToUtf8(33))
cat(intToUtf8(34))
cat(intToUtf8(35))
cat(intToUtf8(36))
cat(intToUtf8(37))
cat(intToUtf8(38))
cat(intToUtf8(39))
`?`=cat;`!`=intToUtf8;?!40
`?`=cat;`!`=intToUtf8;?!41
cat(intToUtf8(42))
cat(intToUtf8(43))
cat(intToUtf8(44))
cat(intToUtf8(45))
cat(intToUtf8(46))
cat(intToUtf8(47))
cat(1-1)
cat(3-2)
cat(1+1)
cat(2+1)
cat(3+1)
cat(4+1)
cat(5+1)
cat(6+1)
cat(7+1)
cat(8+1)
cat(intToUtf8(58))
cat(intToUtf8(59))
cat(intToUtf8(60))
cat(intToUtf8(61))
cat(intToUtf8(62))
cat(intToUtf8(63))
cat(intToUtf8(64))
cat(LETTERS[1])
cat(LETTERS[2])
cat(LETTERS[3])
cat(LETTERS[4])
cat(toupper("e"))
cat(LETTERS[6])
cat(LETTERS[7])
cat(LETTERS[8])
cat(LETTERS[9])
cat(LETTERS[10])
cat(LETTERS[11])
cat(toupper("l"))
cat(LETTERS[13])
cat(LETTERS[14])
cat(LETTERS[15])
cat(LETTERS[16])
cat(LETTERS[17])
cat(toupper("r"))
cat(toupper("s"))
cat(toupper("t"))
cat(LETTERS[21])
cat(LETTERS[22])
cat(LETTERS[23])
cat(LETTERS[24])
cat(LETTERS[25])
cat(LETTERS[26])
cat(intToUtf8(91))
cat(intToUtf8(92))
cat(intToUtf8(93))
cat(intToUtf8(94))
cat(intToUtf8(95))
cat(intToUtf8(96))
write(letters[1],'')
cat(letters[2])
as.name(letters[3])
cat(letters[4])
cat(intToUtf8(101))
cat(letters[6])
cat(letters[7])
cat(letters[8])
cat(letters[9])
cat(letters[10])
cat(letters[11])
cat(intToUtf8(108))
cat(letters[13])
cat(letters[14])
cat(letters[15])
cat(letters[16])
cat(letters[17])
cat(intToUtf8(114))
cat(tolower("S"))
as.name(rawToChar(as.raw(116)))
cat(letters[21])
cat(letters[22])
cat(letters[23])
cat(letters[24])
cat(letters[25])
cat(letters[26])
cat(intToUtf8(123))
cat(intToUtf8(124))
cat(intToUtf8(125))
cat(intToUtf8(126))

Try it online!