Python 2, 47 bytes

lambda l:l.sort(key=len)or l[1::2][::-1]+l[::2]

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R, 63 48 bytes

function(L)c(rev(o<-L[order(nchar(L))]),o)[!0:1]

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Sort by string lengths, then combine the reversed list with the sorted list, finally, take every 2nd element, starting at 1-based index 1.

Javascript 77 bytes

Takes input as an array of strings, outputs an arrow-sorted array of strings.

s=>s.sort((a,b)=>a.length-b.length).reduce((m,x,i)=>i%2?[...m,x]:[x,...m],[])

Explanation

s => // take input as an array of strings s
 s.sort((a,b)=>a.length-b.length) // sort input by string length
 .reduce( // reduce
 (m,x,i)=>i%2?[...m,x]:[x,...m], // if index is even, stick string x at the end of the memo
 // array, else at the beginning
 [] // memo initialized to empty array
 )

C# (Visual C# Interactive Compiler), 89 bytes

n=>(n=n.OrderBy(x=>x.Length)).Where((a,b)=>b%2>0).Reverse().Concat(n.Where((a,b)=>b%2<1))

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K (oK), 24 bytes

Solution:

x(<#:'x)(|&~w),&w:2!!#x:

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Explanation:

Generate the 6 4 2 0 1 3 5 sequence, use that to index into the ascending lengths of input, and use that to index into the original array:

x(<#:'x)(|&~w),&w:2!!#x: / the solution
 x: / save input as x
 # / count (#) of x
 ! / range 0 to ...
 2! / modulo 2
 w: / save as w
 & / indices where true
 , / join with
 ( ) / do this together
 ~w / not (~) w
 & / indices where true
 | / reverse
 ( ) / do this together
 #:'x / count (#:) of each (') x
 < / indices to sort ascending
x / index into x

Jelly, 9 8 bytes

LÞŒœm"-Ẏ

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LÞŒœṚ;¥/

is also 8 bytes.

Thanks to @EriktheOutgolfer and @JonathanAllan for both offering golfs to save a byte.

Ruby, 51 bytes

->lr=1;l.sort_by!(&:size).mapl[r-=2]

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05AB1E, 6 5 bytes

Saved 1 byte thanks to Kevin Cruijssen

I/O is a list of strings.

Link is modified for newline separated I/O for easier testing.

éι`Rì

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Explanation

é # sort by length ascending
 ι # uninterleave into 2 parts, both sorted ascending
 ` # push the 2 parts separately to the stack
 R # reverse the second part
 ì # and append it to the first

J, 11 bytes

,~`,/@:#&>

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We sort it down first.

Then we reduce the list form right to left, but alternating which side we put the new element on. Done.

PowerShell, 66 bytes

1..($a=$args|sort l*).count|?$_%2|%$a[-$_];$x=,$a[-++$_]+$x;$x

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Takes input via splatting, which manifests on TIO as separate command-line arguments. sorts on the length, stores that into $a, and constructs a range from 1 up to the count of input strings. We then pull out only the odd ones ?$_%2 and feed those into a loop |%.... Each iteration, we put the "last", then the "third from last", and so on onto the pipeline with $a[-$_]. Separately, we also accumulate into $x the "second from last", "fourth from last", etc. Out of the loop and the pipeline is flushed (so those elements are output) and then we output $x. In both instances, the default output gives us newlines between items automatically.

PHP, 144 141 bytes

function($a)usort($a,function($b,$c)return strlen($b)-strlen($c););$e=[];foreach($a as$d)(array_.[unshift,push][++$i%2])($e,$d);return$e;

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-3 bytes thanks to @Ismael Miguel!

MATLAB, 87 bytes

function f(y);[B,I]=sort(cellfun(@(x)length(x),y));yflip(I(1:2:end)),yI(2:2:end)'

Takes input as cell array of strings, outputs column of strings (not sure if that's legal)

> s = 'qweq qwe qw','qweqw','12132132131231231','asdasdasda','qwe','w';
> f(s)
> >> 
> ans =
> 
> 6×1 cell array
> 
> 'qweq qwe qw' 
> 'qweqw' 
> 'qwe' 
> '1234' 
> 'asdasdasda' 
> '12132132131231231'

PS: Thanks Sanchises for pointing to a bug with odd-length inputs

APL (Dyalog Unicode), 18 16 bytes^SBCS

⊢⌷⍨∘⊂∘⍋≢¨×¯1*⍳∘⍴

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-2 bytes thanks to @dzaima

Explanation of this giant composition of function:
(all the ∘ do is compose these functions together)

⊢⌷⍨∘⊂∘⍋≢¨×¯1*⍳∘⍴ ⍝ A giant train of functions. Argument is ⊢
 ⍴ ⍝ The shape of the argument (number of strings)
 ⍳ ⍝ Range 1..shape
 ¯1* ⍝ Calculate -1 to the power of each element in the range
 ⍝ gives ¯1 1 ¯1
 × ⍝ Multiply each with...
 ≢¨ ⍝ The length of the corresponding string
 ⍋ ⍝ Grade: gets the indices for the sorted values
 ⌷ ⍝ Access with these indices into
⊢ ⍝ The argument
 ⊂ ⍝ (enclosed, otherwise the access would be dimension-wise)

¹

APL+WIN, 31 38 bytes

See Adams comment

⊃n[(⍳⍴n)~a],⌽n[a←2×⍳⌊.5×⍴n←n[⍒∊⍴¨n←⎕]]

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Prompts for a nested vector of strings

Retina, 26 bytes

N$`
$.&
*,2,^A`.+
,2,G`.+

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N$`
$.&

Sort the lines in ascending order of length ($.& returns the length of the line).

*,2,^A`.+

Temporarily delete alternate lines and output the remaining lines in reverse order.

,2,G`.+

Keep the only lines that were temporarily deleted and output them.

Gaia, 10 bytes

el∫v:v+2%ụ

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e		| eval as Gaia code (list of strings)
 l∫		| ∫ort by lengths (ascending)
 v:v		| reverse, dup, reverse
 +		| concatenate lists
 2%	| take every other element
 ụ	| join by newlines and output

Japt, 8 bytes

ñÊó g0_w

-3 bytes thanks to Shaggy!

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PowerShell, 49 bytes

$args|sort l*|sort$_.Length*($global:x=-$x*2+1)

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The double distillation.

T-SQL, 84 bytes

Input is a table variable

SELECT a FROM(SELECT*,row_number()over(order by len(a))r
FROM @)x order by(r%2-.5)*r

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Perl 6, 31 bytes

*.sort(&chars).sort:$++%2*$--

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Sort by string length, then by static sequence 0, -1, 0, -3, 0, -5, ...

Javascript 95 Bytes

s=>s.sort((x,y)=>x.length-y.length).reduce((a,e,i)=>i%2?a.push(e):a.unshift(e);return a;,[]);

Red, 116 101 bytes

func[b][sort/compare b func[x y][(length? x)> length? y]collect[forall b[keep take b]keep reverse b]]

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perl 5 (-p0777F/n/ -M5.01), 59 bytes

for$x(sort$b=~y///c-length$a@F)--$}improve this answer