Got to be kidding!

Solution with only one line.

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def bell(n):
    "Got to be kidding!"
    lst = [1] + [0]*(n-1)
    for m in range(n -1):
        save, lst[0]  = lst[0], lst[m]
        for k in range(1, m + 2):
            save, lst[k] = lst[k], lst[k-1] +  save
    return  lst[n-1]

print(len(str(bell(1000))))

To compute the next row only the current row is needed, which saves space. Also, we need not keep any Bell numbers preceding the 1000th.

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def Bell():
    line_1=[1]
    while True:
        yield line_1[0]
        line_2=[line_1[-1]]
        for _ in range(len(line_1)):
            line_2.append(line_1[_]+line_2[_])
        line_1=line_2

for n, b in enumerate(Bell()):
    if n==1000:
        print(len(str(b)))
        break

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def bell_numbers(start, stop):

   t = [[1]]                        ## Initialize the triangle as a two-dimensional array
   c = 1                            ## Bell numbers count
   while c <= stop:
       if c >= start:
           yield t[-1][0]           ## Yield the Bell number of the previous row
       row = [t[-1][-1]]            ## Initialize a new row
       for b in t[-1]:
        row.append(row[-1] + b)  ## Populate the new row
       c += 1                       ## We have found another Bell number
       t.append(row)                ## Append the row to the triangle
   for b in bell_numbers(1, 1000):
       print b

Then I copied the whole number and pasted it in answer box but we can only paste 33 digits and it shows how much we pasted.

$Wolfram Alpha$ to the rescue .

The answer is 10^1927.47991= 29899013356824084214804223538976464839473928098212305047832737888945413625123259596641165872540391578300639147082986964028021802248993382881013411276574829121155811755170830666039838837273971971676782389800810361809319250755399325279656765435255999301529770267107281619733800281695881540007577899106878679451165492535930459233713316342551545242815802367257284852612201081016386308535990145447341800455472334713864080523978960296365736999295932080550928561633025800627524911700149562106895897725047744775812241800937310491797818107578233924187312824632629095993832334781713007323483688294825326897450386817327410532925074613888321264138083842196202242956001314953449497244271843922741908252107652201346933889741070435350690242062001522697855278356012055718392851567813397125419144780476479197990921602015873703820769182603836788465785093563686025690269802153802436873530877006737154523895273029510238745997356292232631282773748762989386003970214423843947094021177989737557020369751561595003372955621411858485959813344799967960196238368337022346946771703060269288691694028444791203978533454759410587065022546491518871238421560825907135885619221776405898771057270555581449229994215739476758785884545723062263992367750091319644861547658472282284005892044371587560711880627741139497818835632120761570174928529697397267899554407350161283097123211048049269727655279783900702416095132827766428865017653366696304131436690232979453876337599721772897049270230544262611264917393374756384152784943607952408782612639220380791445272655004475989064276373713608901650681165467490310898804916827069427310961109285035545084791339423266482359955663377201515204340817580915468489969181643341007197836481461051798995640789292580146918580703759556634019451731530034209189203377522668309771129566108101617727442045637098112678864654309987785463307376544339506878267267349348171320834971956806668304099159992067385998690820326902473886782781499414773179 It has $1928$ digits.